Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II
Nikolaos Kapouleas Peter McGrath
math.DG


Doubling]Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II

N. Kapouleas]Nikolaos Kapouleas
P. McGrath]Peter McGrath

Department of Mathematics, Brown University, Providence,
RI 02912 peter_mcgrath@math.brown.edu nicos@math.brown.edu







In <cit.> 
new closed embedded smooth minimal surfaces in the round three-sphere 𝕊^3(1) were constructed, 
each resembling two parallel copies of the equatorial two-sphere  joined by small catenoidal bridges,  
with the catenoidal bridges concentrating along two parallel circles, 
or the equatorial circle and the poles.  
In this sequel 
we generalize those constructions 
so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, 
with the further option to include bridges at the poles. 
The current constructions follow the Linearized Doubling (LD) methodology developed in <cit.>  
and the LD solutions constructed here can be modified readily for use 
to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides <cit.>. 
In particular they allow us to develop in <cit.> doubling constructions for 
the catenoid in Euclidean three-space, the critical catenoid in the unit ball, 
and the spherical shrinker of the mean curvature flow.  

Unlike in <cit.>, our constructions here 
allow for sequences of minimal surfaces where the catenoidal bridges tend to be “densely distributed”,  
that is do not miss any open set of  in the limit. 
This in particular leads to interesting observations which seem 
to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities 
by concentrating infinitely many catenoidal necks at a point. 

[
    [
    December 30, 2023
=====================










§ INTRODUCTION


ab


 §.§ The general framework

ab


This article is the second one 
in a series in which gluing constructions
for closed embedded minimal surfaces in the round three-sphere ^3(1)
by doubling the equatorial two-sphere  are discussed. 
Doublings of the equatorial two-sphere  are important 
as a test case for developing the doubling methodology and also 
because their area is close to 8π 
(the area of two equatorial two-spheres), 
a feature they share with the celebrated surfaces 
constructed by Lawson in 1970 <cit.>.
The classification of the low area 
closed embedded minimal surfaces in the round three-sphere ^3(1), 
especially of those of area close to 8π 
or less, 
is a natural open question.
This is further motivated by the recent resolutions of 
the Lawson conjecture by Brendle <cit.> 
and the Willmore conjecture by Marques and Neves <cit.> 
where they also characterize the Clifford torus and the equatorial sphere
as the only examples of area ≤2π^2. 
We refer to <cit.> for a survey of existence and uniqueness
results for minimal surfaces in the round three-sphere.

The general idea of doubling constructions by gluing methods 
was proposed and discussed in 
<cit.>.
Gluing methods have been applied extensively
and with great success in Gauge Theories by Donaldson, Taubes, and others.
The particular kind of gluing methods used in this article relate most
closely to the methods developed in <cit.> and <cit.>,
especially as they evolved and were systematized in
<cit.>.
We refer to <cit.> for a general discussion of this gluing methodology 
and to <cit.> for a detailed general discussion of doubling by gluing methods.

Roughly speaking, in such doubling constructions,  
given a minimal surface Σ,
one constructs first a family of smooth, embedded, and approximately minimal, initial surfaces. 
Each initial surface 
consists of two approximately parallel copies of Σ with a number of discs removed and replaced by approximately catenoidal bridges.
Finally, one of the initial surfaces in the family is perturbed to minimality by Partial Differential Equations methods.
Understanding such constructions in full generality seems beyond the immediate horizon at the moment.
In the earliest such construction <cit.> 
where doublings of the Clifford torus are constructed, 
there is so much symmetry imposed, 
that the position of the catenoidal bridges is completely fixed 
and all bridges are identical modulo the symmetries.
Moreover the bridges are uniformly distributed, 
that is when their number is large enough, 
there are bridges located inside any preassigned domain of Σ.
Wiygul <cit.> has extended that construction to multiple doublings 
with more that two copies of the Clifford torus involved (and some less symmetric doublings also), 
where the symmetries do not determine the vertical (that is perpendicular to Σ) 
position of the bridges.

In a previous article <cit.> doubling constructions where 
the horizontal position of the bridges is not determined by the symmetries, 
or there are more than one bridge modulo the symmetries, 
were carried out for the first time.  
To realize such constructions an intermediate step in the construction was introduced.  
In this intermediate step singular solutions of the linearized equation, 
called linearized doubling (LD) solutions, 
are constructed and studied.  
This new approach which we call Linearized Doubling (LD),  
provides a systematic methodology for dealing with the obstructions involved 
and also provides a detailed understanding of the regions further away from the catenoidal bridges.
It can also be generalized to higher dimensions <cit.>. 
In <cit.> the conversion of suitable continuous families of LD solutions on  
into minimal surfaces whose catenoidal bridges “replace” the singularities of the LD solutions was realized in general as part of the LD methodology. 
This reduced the construction of the minimal surfaces to the construction and estimation of the LD solutions, 
which remained however a very difficult problem.  

In <cit.> the only LD solutions which were constructed had their singularities either on two parallel 
circles symmetrically arranged around the equatorial circle of , or 
at the poles and the equatorial circle of .  
This way two (discrete) families of minimal surfaces were obtained, 
a family with the catenoidal bridges concentrating on two parallel circles symmetrically arranged around the equatorial circle of 
and a family where there are two bridges at the poles and the rest concentrate on the equatorial circle of .  



 §.§ Brief discussion of the results

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In this article we study LD solutions with singularities on an arbitrary number (but ≥2) of parallel 
circles of  (subject to the same symmetries as in <cit.>), 
with the option to have singularities at the poles also.  
This leads to the construction of minimal doublings of  with the catenoidal 
bridges replacing the singularities of the LD solutions 
(see Theorems <ref>, <ref>, <ref>, and <ref>)
and thus concentrating along an arbitrary number ≥2 of parallel circles and optionally at the poles.  

In particular we obtain for the first time (unlike in <cit.>) 
sequences of minimal doublings of  where 
the number of the parallel circles tends to infinity 
and therefore the number of bridges contained in any 
fixed in advance open subset of  tends also to infinity, 
that is the catenoidal bridges become “densely distributed” in the limit. 
We observe then the interesting phenomenon 
that the size of the catenoidal bridges on each minimal surface of such a sequence 
tends to become uniform,  
in the sense that 

    lim_→∞ lim_→∞ τ_max/τ_min =1,

where  is the number of the parallel circles,  the number of bridges on each circle, 
and
τ_max/τ_min the ratio of the maximum size over the minimum size of the bridges  
(see Remarks <ref> and <ref>). 
This happens even in the case where nearby catenoidal bridges experience very different geometries, 
as when the bridge at a pole is surrounded by a very large number of bridges on nearby parallel circles.  
This suggests that when there is a very large number of catenoidal bridges close to a point, 
even in asymmetric situations, the sizes of the bridges would tend to become uniform. 
This would imply a negative answer to the very important question on whether embedded minimal surfaces 
with isolated singularities can be constructed by concentrating an infinite number of catenoidal 
bridges in the vicinity of a singular point. 

Finally we remark that the families of LD solutions we construct here find immediate application in <cit.> 
to constructions of complete minimal surfaces in the Euclidean space by doubling the catenoid, 
constructions of free boundary minimal surfaces in the unit ball by doubling the critical catenoid, 
and of self-shrinkers of the mean curvature flow by doubling the spherical self-shrinker: 
Although the LD methodology as applied in <cit.> and in the current paper assumes 
that the minimal surface being doubled possesses a symmetry fixing it pointwise and exchanging 
its sides, as  does, 
it is possible 
<cit.>  
to extend the methodology to the case of asymmetric sides, 
as in the case of the catenoid and the spherical self-shrinker.  
Because the catenoid is conformally isometric to  and the spherical self-shrinker is round 
we can use the LD solutions we derive here with small modifications to carry out the constructions 
in <cit.>. 












 §.§ Outline of the approach

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In this article, as in <cit.>, in order to construct a family of initial surfaces 
one of which is later perturbed to minimality by a fixed point theorem,  
we first construct a family of LD solutions 
(defined in <cit.> and <ref>). 
The LD solutions are converted to 
matched linearized doubling (MLD) solutions 
(defined in <cit.> and <ref>),  
which are then converted to initial surfaces. 
In both articles the construction and estimation of the LD solutions relies 
on the rotational invariance of . 
Each LD solution φ is related to a solution ϕ obtained by averaging φ
on the circles of latitude  
(see <cit.> and <ref>). 
ϕ then belongs to a class of rotationally invariant solutions which in this article 
we call 
rotationally invariant linearized doubling (RLD) solutions (see  <ref>). 
The RLD solutions studied in this article are rather more complicated compared to the ones studied in <cit.>. 
Much of the progress achieved involves their detailed understanding. 
We summarize schematically the main steps in the construction in the following diagram 
(see Remark <ref> for a more detailed outline): 

 
We discuss now the main innovations of this paper. 
A large part of the effort is in understanding and estimating in detail the RLD solutions. 
Achieving this is helped by the observation that the class of LD solutions is invariant 
under conformal changes of the intrinsic metric of the surface and therefore RLD solutions can 
be considered as defined on the flat cylinder instead of the round sphere. 
We also employ a dimensionless flux F^ϕ, 
which amounts to the logarithmic derivative of the RLD solution ϕ, 
to carefully study the RLD solutions. 
Such a quantity was used also in <cit.>, 
but here we study F^ϕ much more carefully 
and we establish that it satisfies a Riccati differential equation and has several useful monotonicity properties.  

The fact that unlike in <cit.> we are dealing with situations where there are more than one (modulo the symmetries) circles of latitude where 
the catenoidal bridges concentrate, makes the balancing and unbalancing questions for the RLD solutions harder to study. 
The fluxes F^ϕ are the main tool which allows us to study these questions systematically. 
We also have developed a careful and systematic approach to study the parameters related to unbalancing (see <ref>). 



The current article incorporates various improvements in the application of the LD methodology: 
First, 
we estimate our LD solutions on the flat cylinder instead of the equatorial two-sphere as in <cit.>  
by making use of the conformal invariance of the LD solutions. 
The uniformity and further symmetries of the cylinder allow us to obtain stronger estimates.  
Second, we improve the analysis of the relation between the LD solutions and the corresponding RLD solutions. 
In particular, the decomposition 
Φ=++ in <ref> of an LD solution Φ into a singular part , 
a rotationally invariant part , and an error term , 
is an improvement over the two decompositions <cit.> it replaces. 
Finally, we employ a simplified definition for [L] in <ref>. 



 §.§ Organization of the presentation

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In Section <ref>, we review definitions and notation from <cit.> 
relating to the elementary geometry of the objects we are interested in 
and catalog a useful conformal diffeomorphism (<ref>) between the cylinder and the twice-punctured two-sphere.
We also discuss some special rotationally invariant solutions and Green's functions for the linearized equation. 

The main new features in this paper which refine the approach in <cit.> take place in Sections <ref> and <ref>.  
In Section <ref>, we define in <ref> a class of rotationally invariant solutions of the linearized equation (RLD solutions), 
establish appropriate criteria for their existence and uniqueness (see <ref>), 
and prove estimates governing their geometry in <ref>, 
and behavior under small perturbations of initial data in <ref>.

In Section <ref>, we more generally study linearized doubling solutions (LD solutions) on the cylinder which have prescribed logarithmic singularities at L. 
More precisely in Lemma <ref> we convert RLD solutions  to corresponding LD solutions Φ. 
We introduce then in <ref> the decomposition Φ=++ of an LD solution Φ into a singular part , 
a rotationally invariant part , and an error term .  
Much of the remaining work in the section is devoted to estimating  in <ref>.

In Section <ref>,
we convert the LD solutions we have constructed in Section <ref>
to MLD solutions which satisfy the appropriate linear and nonlinear matching conditions.  
Using the earlier estimates, we prove in <ref> the estimates we need for these MLD solutions.  
In Section <ref>, 
we convert these families of MLD solutions into families of smooth initial surfaces with small mean curvature.
by recalling work from <cit.>, 
and we then apply a fixed point theorem to perturbations of the families of initial surfaces constructed in Section <ref> 
to construct our minimal surfaces.  
Finally in Section <ref> we modify the construction to include catenoidal bridges at the poles or on the equatorial circle of  or both.





 §.§ Acknowledgments

ab


The authors would like to thank Richard Schoen for his continuous support and interest in the results of this article. 
NK would like to thank the Mathematics Department and the MRC at Stanford University
for providing a stimulating mathematical environment and generous financial support during Spring 2016.
NK was also partially supported by NSF grant DMS-1405537.  

















§ ELEMENTARY GEOMETRY AND NOTATION





 §.§ Hölder norms and cut-off functions


ab



We will find the following notation useful. 


We write a∼_c b to mean that 
a,b∈ are nonzero of the same sign, 
c∈(1,∞), 
and 1/c≤a/b≤ c. 


We use the standard notation u: C^r,β( Ω,g ) 
to denote the standard C^r,β-norm of a function or more generally
tensor field u on a domain Ω equipped with a Riemannian metric g.
Actually the definition is completely standard only when β=0
because then we just use the covariant derivatives and take a supremum
norm when they are measured by g.
When β0 we have to use parallel transport along geodesic segments 
connecting any two points of small enough distance
and this may be a complication if small enough geodesic balls are not convex.
In this paper we take care to avoid situations where such a complication
may arise and so we will not discuss this issue further.

We adopt the following notation from <cit.> for weighted Hölder norms.  


Assuming that Ω is a domain inside a manifold,
g is a Riemannian metric on the manifold, 
r∈_0, 
β∈[0,1), u∈ C^r,β_loc(Ω) 
or more generally u is a C^k,β_loc tensor field 
(section of a vector bundle) on Ω, 
ρ,f:Ω→(0,∞) are given functions, 
and that the injectivity radius in the manifold around each point x in the metric ρ^-2(x) g
is at least 1/10,
we define

    u: C^r,β ( Ω,ρ,g,f):=
    sup_x∈Ω u:C^r,β(Ω∩ B_x, ρ^-2(x) g) /f(x) ,

where B_x is a geodesic ball centered at x and of radius 1/100 in the metric ρ^-2(x) g.
For simplicity we may omit any of β, ρ, or f, 
when β=0, ρ≡1, or f≡1, respectively.


f can be thought of as a “weight” function because f(x) controls the size of u in the vicinity of
the point x.
ρ can be thought of as a function which determines the “natural scale” ρ(x)
at the vicinity of each point x.
Note that if u scales nontrivially we can modify appropriately f by multiplying by the appropriate 
power of ρ.  Observe from the definition the following multiplicative property: 

    u_1 u_2   : C^k,β(Ω,ρ,g,  f_1 f_2   )≤
    C(k)   u_1   : C^k,β(Ω,ρ,g,  f_1   )    u_2   : C^k,β(Ω,ρ,g,  f_2   ).


Our arguments will require extensive use of cut-off functions, and it will be helpful to adopt the following. 

 
We fix a smooth function Ψ:→[0,1] with the following properties:


  * Ψ is nondecreasing.


  * Ψ≡1 on [1,∞) and Ψ≡0 on (-∞,-1].


  * Ψ-1/2 is an odd function.



Given a,b∈ with a b,
we define smooth functions
[a,b]:→[0,1]
by

    [a,b]:=Ψ∘ L_a,b,

where L_a,b:→ is the linear function defined by the requirements L(a)=-3 and L(b)=3.

Clearly then [a,b] has the following properties:


  * [a,b] is weakly monotone.


  * [a,b]=1 on a neighborhood of b and 
[a,b]=0 on a neighborhood of a.


  * [a,b]+[b,a]=1 on .



Suppose now we have two sections f_0,f_1 of some vector bundle over some domain Ω.
(A special case is when the vector bundle is trivial and f_0,f_1 real-valued functions).
Suppose we also have some real-valued function d defined on Ω.
We define a new section 

    [a,b;d  ](f_0,f_1):=
    [a,b  ]∘ d   f_1
    +
    [b,a]∘ d   f_0.

Note that
[a,b;d  ](f_0,f_1)
is then a section which depends linearly on the pair (f_0,f_1)
and transits from f_0
on Ω_a to f_1 on Ω_b,
where Ω_a and Ω_b are subsets of Ω which contain
d^-1(a) and d^-1(b) respectively,
and are defined by

    Ω_a=d^-1((-∞,a+1/3(b-a))),
        Ω_b=d^-1((b-1/3(b-a),∞)),

when a<b, and 

    Ω_a=d^-1((a-1/3(a-b),∞)),
        Ω_b=d^-1((-∞,b+1/3(a-b))),

when b<a.
Clearly if f_0,f_1, and d are smooth then
[a,b;d  ](f_0,f_1)
is also smooth.







 §.§ The parametrizations Θ and  and the coordinates (,,) and (, θ)

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We consider now the unit three-sphere ^3(1)⊂^4.
We denote by (x_1,x_2,x_3,x_4) the standard coordinates of ^4
and we define by

    :=^3(1)∩{x_4=0}

an equatorial two-sphere in ^3(1).
To facilitate the discussion we fix spherical coordinates (,,) on ^3(1)
by defining a map Θ:^3→^3(1) by

    Θ(,,) = 
    (coscoscos, cossincos, sincos,sin).

Note that in the above notation we can think of  as the geographic latitude on 
and of  as the geographic longitude.
We will also refer to

    _0:= ∩{x_3=0}=
    Θ({==0}),
    
    
    p_N:=(0,0,1,0)=Θ(π/2,,0),
    
    
    p_S:=(0,0,-1,0)=Θ(-π/2,,0),

as the equator circle, the North pole, and the South pole of  respectively.

Clearly, the standard metric of ^3(1) is given in the coordinates of (<ref>)
by

    Θ^* g = cos^2 (   d^2 +cos^2 d^2  )  + d^2.

Finally we define a nearest-point projection 
:^3(1)∖{(0, 0, 0, ± 1)}→
by

    (x_1,x_2,x_3,x_4)=
    1/|(x_1,x_2,x_3,0)|
    (x_1,x_2,x_3,0).

Clearly we have

    ∘Θ(,,)
    =
    Θ(,,0).


The study of the RLD and LD solutions can be simplified by the observation that ∖{p_N, p_S} is conformally equivalent to a flat cylinder ×^1. 
To be precise, let

    : = ×^1

be the cylinder endowed with the flat product metric χ,  
where 

    χ: = d^2 + dθ^2

and (,θ) are the standard coordinates on . 
Consider the map :  →∖{p_N, p_S} defined by 

    (, θ) = ( cosθ, sinθ, tanh ) .

Clearly  is a diffeomorphism and from now on we will use it to identify 
∖{p_N, p_S} with .  
 and θ can then be considered as coordinates on 
∖{p_N, p_S} and by (<ref>)
we have 

    sin = tanh ,     cos = ,      = log1+sin/cos,    and   = θ.

Straightforward computations show 

    g = (^2 ) χ,  
        d /d  = 1/cos, 
        d/d = .


We finally introduce some convenient notation.



We use the shorthand notation {∈ (a, b)} to denote the annular region 
{ (, θ) ⊂: ∈ (a, b)} and similar abbreviations with regard to other types of intervals. 




For X a subset of  and h a Riemannian metric on  or on ∖{p_N, p_S}, 
we write ^h_X for the distance function from X, with respect to h.  We define a tubular neighborhood of X by

    D^h_X(δ):={p∈:^h_X(p)<δ}.
 
If X is finite we just enumerate its points in both cases. 
For example, ^g_q(p) is the geodesic distance between p and q and
D^g_q(δ) is the geodesic disc in  of center q and radius δ.








 §.§ The symmetries and the configurations


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To study the symmetries of the parametrization Θ,
we first define
reflections
of its domain 
, _c, :=_0, and ,
and translations
_c,
where c∈, by 

    (,,):=(-,,),    _c(,,)   :=(,2c-,),    (,,):=(,,-),
    
    _c(,,)   :=(,+c,).



We also define corresponding
reflections
, _c, :=_0, and 
and rotations _c,
of ^4 ⊃^3(1) by

    (x_1,x_2,x_3,x_4):=   (x_1,x_2,-x_3,x_4),
    
    (x_1,x_2,x_3,x_4):=   (x_1,-x_2,x_3,x_4),
    
    (x_1,x_2,x_3,x_4):=   (x_1,x_2,x_3,-x_4),
    
    _c(x_1,x_2,x_3,x_4):=   (x_1cos 2c + x_2sin 2c ,  x_1 sin 2c- x_2 cos 2c,x_3, x_4)
    
    _c(x_1,x_2,x_3,x_4):=   (x_1cos c - x_2sin c, x_1sin c + x_2cos c,x_3, x_4).


Note that , , , and _c
are reflections with respect to the 3-planes
{x_3=0}, {x_2=0}, {x_4=0}, and _c({x_2=0}),
respectively.
 exchanges the two sides of  which it fixes pointwise.
Clearly _2π is the identity map.
We record the symmetries of Θ in the following lemma:



Θ restricted to

    :=
    (-π/2,π/2) ××(-π/2,π/2)

is a covering map onto 
^3(1)∖{x_1=x_2=0}.
Moreover the following hold:


  * The group of covering transformations is generated by
_2π.


  * ∘Θ=Θ∘,
_c∘Θ=Θ∘_c,
∘Θ=Θ∘,
and
_c∘Θ=Θ∘_c.








The symmetry group of our constructions depends on a large number m∈
which we assume now fixed.
We define =[m]⊂ to be the union of m meridians symmetrically arranged: 

    =[m]:=Θ({(,,0):∈[-π/2,π/2],=2π i/m, i∈}).


 

We denote by  and  the groups of 
isometries of ^3(1) and  respectively
which fix [m] as a set.


Clearly  is a finite group and is generated by the reflections
, ,  and _π/m, that is  

    = ⟨, , , _π/m⟩,

and moreover  can be identified with the subgroup of  generated by , and _π/m.
The configuration of our constructions also depends on a number k∈ whose values are restricted in terms of m (see <ref> below).



Given ∈ [0, ∞), we define 

    L_par[] = { (, θ) ∈ :  = ±}.

Furthermore, given 
:= (_1, …, _k)∈^k_+ 
such that 
0 <_1<⋯<_k < ∞, 
we define 
 
    L_par[] = ⋃_i=1^k L_par[_i].

Finally given also 
a domain Ω⊂, 
we will denote by Ω^ the 
“subdivision” of Ω by L_par[]: 
More precisely Ω^ is the abstract surface 
which is the disjoint union of the Ω∩ A's,  
where A is the closure of any connected component 
(a disk or an annulus) of ∖ L_par[].  
Clearly functions on Ω can be thought of as functions on Ω^ as well. 


Note for example that a function defined on Ω which 
is in C^∞(Ω^) is also in C^0(Ω) but not necessarily 
in C^1(Ω); it is “piecewise smooth” on Ω.



For m as in <ref> and 
 and  as in <ref>, 
we define  

    L[ ; m]  : =    L_mer[m]∩ L_par[ ]  =  (,0),
    
     
    L = L[; m] :=    L_mer[m]∩ L_par[] = ⋃_i=1^k L[_i;m].

For i ∈{1,...,k} we define p_i:=(_i,0), L_i=L[_i;m], 
and given also a -symmetric function τ:L[; m]→, 
τ_i := τ(p_i).  









 §.§ The linearized equation and the cylinder
 

ab


A major step in the construction is to estimate solutions of the Jacobi equation u  = 0, where

    = Δ_g + |A|^2 + (ν, ν)= Δ_g+ 2.

We define also a version of ' useful 
in the cylindrical picture by 

    _χ := Δ_χ + 2^2  = ^2  ', 
         where     Δ_χ := ∂^2/∂^2 +  ∂^2/∂θ^2.
 

It will be easier to state some of our estimates if we use a scaled metric  on  and corresponding scaled coordinates ( ,) defined by 

    :=m^2 g_,
        =m,
        =m.

In the same fashion, we define a scaled metric  on  and scaled coordinates
  ( ,   )
defined by

    := m^2 χ, 
         = m , 
         = m θ.

We also define corresponding scaled linear operators

    _ : = Δ_ + 2m^-2 = m^-2, 
        _: = Δ_+2m^-2^2  = m^-2_χ.


For future reference, we record the following global parametrization of a catenoid with unit waist size:

    _cat(, θ) = ( coshcosθ, coshsinθ, ).

For τ>0,  τ_cat parametrizes the catenoid with waist size τ.




Let ∈_+.  We define a shifted coordinate =   [] by 

    :=  - m .
 

 


For convenience we define 

    δ := 1/(9m).

Given ∈ [0, ∞), we have 
nested open sets D^χ_L[;m](3δ) ⊂Ω'[;m]⊂Ω[;m] where 
 
    Ω[; m] : = D^χ_L_par[](3/m),    Ω'[; m] := 
     D^χ_L_par[](2/m).

 


 
 
Given a domain Ω⊂ which is invariant under the coordinate reflection through some ∈, 
we define a reflection operator _: C^0(Ω)→ C^0(Ω) 
by 

    _ u(+', θ) = u(- ', θ),   for   (+ ', θ) ∈Ω

and an antisymmetry operator _: C^0(Ω)→ C^0(Ω) by 

    _ u = u - _ u.
 






Let ∈(3/m, ∞).  The following hold:


  * For all u, v ∈ C^0_(Ω[; m]),  

    _[ u v] = u  _ [v] + _ [u]  _ [v].
 


  * For all u ∈ C^2_(Ω[; m]),

    _[ _χ u]    = _χ[ _ u] + 2_ [^2 ]  _ [u], 
    _[ _ u]    = _[ _ u] + 2m^-2_ [^2 ]  _ [u].



  * 
 
  * Let Ω⊂ be a domain.  Then 
^2  : C^j( Ω , χ) ≤ C(j) ^2  : C^0(Ω, χ). 

  * _[ ^2 ] : C^j( Ω[; m], χ )≤C(j)/m^2  : C^0( Ω[; m], χ ).




(i) follows from a straightforward computation, and (ii) follows from (i) and a similar computation, using the fact that Δ commutes with _.  

For (iii).(a), observe that for each j≥ 1,  ∂^j( ^2) is a polynomial expression in ^2 and tanh each term of which contains a factor of ^2.  (iii).(b) is a discrete version of (iii).(a) which follows from the mean value theorem.  More precisely, fix ∈_+ and let ∈ (-3, 3), where =   [] is as in <ref>.   By the mean value theorem, there exists ' ∈ (- , ) such that

    _[ ^2 ](  +/m)    = ^2(  +/m) - ^2(   - /m)
       = 4  /m^2(  +'/m)tanh(  +'/m).

Estimating the right hand side of the preceding, it follows that

    _[ ^2 ] : C^0( Ω[; m], χ )≤C/m^2   : C^0( Ω[; m], χ ).
 
Estimates on higher order derivatives follow from the mean value theorem in a similar way.





 §.§ Rotationally invariant functions


ab

We call a function defined on a domain of  which depends only on the coordinate  a  rotationally invariant function.  
The linearized equation _χϕ = 0 amounts to an ODE when ϕ is rotationally invariant. 
Motivated by this, we introduce some notation to simplify the presentation:  



Consider a function space X consisting of functions defined on a domain
Ω⊂.
If Ω is invariant under the action of  (recall <ref>),
we use a subscript “” to denote the subspace X_⊂ X
consisting of those functions  f∈ X which are covariant under the action of .
If Ω is a union of parallel circles, we use a subscript “” to denote 
the subspace of functions X_ consisting of 
rotationally invariant functions, which therefore depend only on .
If moreover Ω is invariant under reflection with respect to { = 0},
we use a subscript “||” to denote 
the subspace of functions X_||= X_∩ X_
consisting of those functions which depend only on ||.


For example, we have
C^0_||()⊂ C^0_() ⊂ C^0()
and C^0_||() ⊂ C^0_(),
but C^0_() is not a subset of C^0_().
We also have the following. 



Given a function φ on some domain Ω⊂, we define
a rotationally invariant function φ_ on 
the union Ω' of the parallel circles on which φ is integrable
(whether contained in Ω or not),
by requesting that on each such circle C, 

    . φ_|_C
    :=_Cφ.

We also define φ_ on Ω∩Ω' by

    φ_:=φ-φ_.






If Ω⊂ is a domain and u ∈ C^0_(Ω) has one-sided partial derivatives at =, 
then we denote these partial derivatives by using the notation

    ∂_+ u() :=. ∂ u  /∂|_=+,
            ∂_-  u() :=-. ∂ u  /∂|_=-.

If u is C^1, we use the abbreviation u := ∂ u/∂.  In that case, ∂ u = ∂_+u = -∂_-u.  


If ϕ∈ C^∞_(), the equation _χϕ  =0 (recall <ref>) is equivalent to 

    d^2ϕ/d^2 + 2^2  ϕ = 0.



Define rotationally invariant functions ∈ C^∞_||() and ∈ C^∞_() by

    () = 1-tanh,   () = tanh.





 and  are even and odd in  respectively and satisfy _χ = 0 and _χ = 0.   is strictly decreasing on [0, ∞) and has a unique root _root∈ (0, ∞).   is strictly increasing.  The Wronskian W[ , ] satisfies

    W[, ](): = ()∂() - ∂() () = 1.



Straightforward calculation using Definition <ref> and (<ref>).  



When written in  coordinates (recall (<ref>)),  and   satisfy

    =sin,
        =
    1- sin log1+sin/cos,

and therefore our Definition 
<ref> is equivalent to <cit.>. 



It will be helpful to introduce the following auxiliary ODE solutions:


Given ,∈, and ∈_+ we define

    = [,;]∈     
    C^∞_(  {∈[0,∞)}  )
    ⋂
    C^0_||(   ),
    
    =[;]∈     
    C^∞_( {∈[,∞)} )
    ⋂ 
    C^∞_(  {∈(0,]} )
    ⋂ 
    C^0_||( ),

by requesting they satisfy the initial data 

    ()=,
        ∂ () =   ,
        ()=0,
        ∂_+()=∂_-()=m ,

and the ODEs _=0 on
{∈[0,∞)}, 
and _=0 on {∈[,∞)} and on
{∈[0,]}.




Note that  depends linearly on the pair (, )∈^2 and  depends linearly on ∈.  




Let ∈(3/m, ∞). The following estimates hold (recall <ref>).  


  * [1, 0 ;] -1   :   C_^j(  Ω[; m] ,  )  ≤  C(j)/m^2.


  * [1;] - |   |    :   C_^j(  Ω[; m] ∖[]  ,  )   ≤   C(j)/m^2.


  * _ [1, 0 ;]   : C_^j(  Ω[; m]  ,   )  ≤  C(j)/m^3.


  * _ [1;]   :   C_^j(  Ω[; m] ∖[]   ,  )   ≤   C(j)/m^3.



Denote in this proof = [1, 0; ].  In -coordinates (where =   [ ] is as in <ref>), the equation _ =0  is equivalent to

    ∂^2_   + 2/m^2^2( /m+ )  = 0.

Since ^2t is decreasing on (0, ∞), it is easy to see that for m≥ 6, >0 on Ω[; m], hence 

    ∂^2_   + 2/m^2 > 0   on  Ω[; m].
 
Integrating this differential inequality and matching the initial data implies

    1 -  : C^0_(Ω[ ;m],  )≤ 1 -  cos( √(2)/m ) : C^0_(Ω[ ;m],  )≤C/m^2.

In conjunction with (<ref>), the C^0 bound above implies |  ∂^2_  | < C/m^2 on Ω[, m].  Integrating this bound with respect to  implies | ∂_  |< C/m^2.  Together, these bounds imply the C^2 bound in (i).  Higher derivative bounds follow from the C^2 bound by differentiating (<ref>).  

For (ii), denote = [1; ].  Arguing as above, we have on Ω[;m]∖[]

    ∂^2_   + 2/m^2^2( /m+ )  = 0   and  ∂^2_   + 2/m^2 > 0.

A similar comparison argument establishes that

    -|   | : C^0_ ( Ω[ ; m]∖[] ,  )   ≤m/√(2)sin( √(2)/m  |    | )-|  | : C^0_( Ω[ ; m]∖[_i] ,  )
       ≤C/m^2.

The C^0 bound above implies | ∂^2_  | < C/m^2.
For t∈ (-3, 3), we find by integrating this bound that

    | ∂_  (   ) - ∂_  |     | |
    = | ∂_  (t) -  ∂_  (0)| 
    = | ∫_0^t∂^2_    d  | ≤C/m^2.

This with the preceding bounds implies the C^2 bound in (ii).  Estimates on the higher derivatives follow by differentiating (<ref>).

 By Lemma <ref>.(ii), _ _i satisfies the equation

    ∂^2_  _ + 2/m^2^2 ( /m+ ) _ + 2/m^2_[ ^2 ] ( /m+ _i) _  = 0.

It follows from (i) that 
_   :   C_^2(  Ω[; m] ,  )  ≤  C/m^2.  Using this bound and the estimate on _[ ^2 ] from Lemma <ref>.(iii) in (<ref>), we find | ∂^2_  _ | < C/m^3 on Ω[;m].  Integrating this bound twice with respect to  twice and using the fact that ∂_ _ () = _ () =0 establishes the C^2 bounds in (iii); as before, estimates on the higher derivatives follow from differentiating (<ref>).  The proof of (iv) is almost exactly the same as the proof of (iii), so we omit it.











 §.§ Green's functions and the Green's function for _χ

ab




Let (Σ^2, g) be Riemannian, V∈ C^∞(Σ), and p∈Σ.  If there exists a domain Ω⊂Σ containing p and G_p ∈ C^∞( Ω∖{p}) satisfying 


  * ( Δ_g + V) G_p= 0 

  * For some neighborhood Ω'⊂Ω, 
G_p-log∘^g_p is bounded on Ω'∖{p},

we say  G_p is a Green's function for Δ_g + V centered at p. 


For any (Σ, g), p, and V as above, standard theory guarantees the existence of a Green's function for Δ_g + V with center at p.



Suppose G_p and _p are Green's functions for Δ_g + V, where  p, Ω, V, and (Σ, g) are as in Definition <ref>.  Then G_p - _p has a unique extension to C^∞(Ω).


In this proof, we denote = Δ_g + V.
Definition <ref>.(ii) implies that u: = G_p - _p satisfies u : C^0( Ω∖{ p } , g) ≤ C.  Since u = 0, the C^0 bound and Schauder estimates imply that

    u : C^j( Ω∖{ p } , g) ≤ C(j).


By standard regularity theory, to prove the existence of the extension,  it suffices to prove that u solves u = 0 weakly in Ω.  To this end, let φ∈ C^∞_c(Ω).  Compute

    ∫_Ω u φ  dμ_g 
       =  lim_ϵ↘ 0∫_Ω∖ D^g_p(ϵ)φ u   dμ_g + lim_ϵ↘ 0∫_∂ D^g_p(ϵ) u ∂φ/∂η - φ∂ u /∂η  dμ_g
    = 0,

where the last equality follows from (<ref>) and that  u = 0.




Given two Green's functions G_p, _p as in <ref>, we abuse notation by writing G_p -_p for the smooth extension to C^∞(Ω) of  G_p - _p, whose existence and uniqueness is asserted in Lemma <ref>.


For future reference, we recall <cit.> basic properties of the Green's function used in <cit.> to construct initial surfaces. 


There is a function 
G^^2∈ C^∞((0,π))
uniquely characterized by (i) and (ii) 
and moreover satisfying (iii-vii) below.
We denote by  the standard coordinate of ^+.


  * For small  we have G^^2()=(1+O(^2)) log.


  * For each p∈
we have (G^^2∘^g_p) =0
where
G^^2∘^g_p∈ C^∞(∖{p,-p})
(recall <ref>).


  * G^^2∘^g_p_N= (log 2-1)  + (recall <ref>).


  * G^^2(r)  =   1+  cos r  ( -1+log2sin r /1+cos r ). 


  * ∂ G^^2/∂ r (r)   =   -sin r  log2sin r /1+cos r 
+ 1/sin r + sin r cos r /1+cos r.  


  * G^^2 - 
 cos r  log 
: C^j( 
(0,1) , 
r, dr^2,r^2 ) ≤  C(j)   . 


  * G^^2 
: C^j( 
(0,1) , 
r, dr^2,  |log r|  ) ≤  C(j)   . 



It will also be helpful to have Green's functions G_p for _χ well adapted to the cylinder.


Given p = (, θ)∈, there exists G_p ∈ C^∞(D^χ_p(1/2)∖{p}) satisfying: 


  * _χ G_p = 0 on D^χ_p(1/2)∖{p}. 


  * For q near p, G_p(q) = log r + O(r^2|  log r|), where r(q) = ^χ_p(q).


  * _ G_p : C^j(D^χ_p(1/2) ∖{ p }, r ,χ ,r )≤ C(j). 



As an auxiliary step, we consider solutions of the equation

    Δ_χ u + 2(^2) u = 0.

Let  r = ^χ_p be the polar radius on D^χ_p(1/2).  When u = u(r) is radial, (<ref>) is equivalent to the ODE

    d^2 u/d r^2 + 1/rd  u/d r + 2(^2)   u = 0.

The solution space to (<ref>) is spanned by

    { J_0(√(2)()  r),  Y_0(√(2)()  r)},
 
where J_0 and Y_0 are the Bessel functions (cf. <cit.>) defined by 

    J_0(x)    = ∑_j = 0^∞(-1)^j/4^j (j!)^2 x^2j, 
     
    Y_0(x)    = 2/π{(logx/2+ γ) J_0(x) + ∑_j=1^∞ (-1)^j+1(∑_l=1^j1/l)x^2j/4^j (j!)^2},

and γ is the Euler constant.  
By a short computation, the function G'_p ∈ C^∞( D^χ_p(1/2) ∖{ p }) defined by

    G'_p :=  π/2 Y_0( √(2)()  r) - (γ+log/√(2)) J_0(√(2)()  r)

satisfies

    Δ_χ G'_p + 2(^2 )   G'_p =0   and   G'_p= log r + O(r^2| log r|).

Let w_p ∈ C^2, β(D^χ_p(1/2)) be the unique solution of the Dirichlet problem

    _χ w_p    = 2(^2  - ^2 )G'_p    on   D^χ_p(1/2)
    
    w_p    = 0 on  ∂ D^χ_p(1/2)

and define (recall Definition <ref>)

    w'_p: = w_p + [ -w_p(p), ∂ w_p/∂( p); ].

Finally, define

    G_p = G'_p + w'_p.

By combining (<ref>), (<ref>), and (<ref>) we find _χ G_p = 0, which yields (i). 

Now note that the right hand side of (<ref>) is in C^0, β(D^χ_p(1/2)) for any 0< β < 1.  It follows that w'_p ∈ C^2, β(D^χ_p(1/2)).  In light of (<ref>), w'_p(p) = .∇ w'_p|_p=0.  Therefore, the second order Taylor series for w'_p and the Schauder estimates imply

    w'_p : C^j(D^χ_p(1/2))∖{ p}, r, χ, r)≤ C(j).


(ii) then follows from combining (<ref>), (<ref>), and (<ref>). 

Note from (<ref>) that _ G_p =  _ w'_p.  (iii) then follows from this and (<ref>).



The estimates in Section <ref>, and in particular the decomposition in <ref>, are adapted to the Green's functions G_p, for p∈ L.  
Since for such p, _p is defined (recall  <ref>.(ii)) in terms of the Green's function G^^2∘^g_p, the following lemma will be useful.


 Fix ∈ and let p = (, 0)∈.  We have (recall Convention <ref>)

    ( G_p - G^^2∘^g_p)(p) = -log,     
    d_p(G_p - G^^2∘^g_p)(p) = 1/2tanh    d.



Let q = (, 0), where  is close to .  For convenience, in this proof denote r = ^χ_p(q).  Recalling (<ref>), we have

    ^g_p(q) = | ∫_^ t dt| 
    = |  (- ) + 1/2( - )^2 ( - tanh) + O( (-)^3)|
    
    =(   )r ( 1- 1/2(- )tanh + O(( - )^2)).
 

Consequently, recalling from <ref>.(i) that G^^2(t) = log t + O(t^2 |log t|) for small positive t, we have

    G^^2∘^g_p (q) = log + log r - 1/2 (-)tanh + O(r^2| log r|).


By Lemma <ref>.(ii),  G_p(q)= log r + O(r^2|log r|) and hence

    ( G_p - G^^2∘^g_p)(q) =- log + 1/2(-)tanh + O(r^2 |log r|).

The conclusion then easily follows from (<ref>).













§ ROTATIONALLY INVARIANT SOLUTIONS





 §.§ Basic facts and notation
 
ab


We will estimate our LD solutions by comparing with rotationally invariant solutions.  It will therefore be useful to define a class of rotationally invariant solutions of the linearized equation.  We begin with some notation. 






Let ^ = { (a_i)_i∈: a_i ∈} and _+^ = {(a_i)_i∈: a_i∈, a_i> 0}.  

For any k∈, we identify ^k with a subspace of ^ by the map

    (a_1, …, a_k) ↦ (a_1, …, a_k, 0, 0, …).

We consider the normed space (ℓ^1(^), |· |_ℓ^1) defined by

    ℓ^1(^) = { = (a_i)_i∈∈^ : ∑_i=1^∞ |a_i|< ∞},   
    | |_ℓ^1 = ∑_i=1^∞ |a_i|.



 


If = (σ_i)_i∈∈ℓ^1( ^) and = (ξ_i)_i∈∈ℓ^∞( ^) satisfies | |_ℓ^∞ < 1/10 and satisfy

    e^σ_i = F_i+1+ + F_i+1-/F_i+ + F_i-,     ξ_i = F_i+ - F_i-/F_i++F_i-   i∈,

for some positive numbers F_i±, i∈, then note that for any 1 ≤ j ≤ i < ∞, 

    F_i+ = 1  + ξ_i /1   + ξ_j( e^∑_l=j^i-1σ_l) F_j+   = 1  + ξ_i /1   - ξ_j( e^∑_l=j^i-1σ_l) F_j-,
    
     
    F_i- = 1  - ξ_i /1   + ξ_j( e^∑_l=j^i-1σ_l) F_j+   = 1  - ξ_i /1   - ξ_j( e^∑_l=j^i-1σ_l) F_j-,

and therefore 

    sup{ F_i±}_i∈∼_exp(||_ℓ^1+3||_ℓ^∞)inf{ F_i±}_i∈  .







 §.§ RLD solutions and the scale invariant flux

ab




If  Ω = {∈ (a, b)}⊂, ϕ∈ C^0_(Ω), and  ϕ is piecewise smooth and positive on Ω, 
we define F^ϕ_±:(a, b)→ by (recall <ref>) 

    F^ϕ_±() = ∂_±ϕ()/ϕ().
 
 



Note that F^ϕ_± = F^cϕ_±  ∀ c∈_+.  Also, if ϕ is C^1 at =, then F^ϕ_+() = - F^ϕ_-().  



We say ϕ∈ C^0_||( ) is a rotationally invariant linearized doubling (RLD) solution if (recall <ref>) 


  * ϕ >0. 

  * There is k∈ and ^ϕ∈^k_+ as in <ref> 
such that 
ϕ∈ C^∞_|| ( ^^ϕ ) 
and _χϕ =0 on ^^ϕ.  

  * For i=1, …, k, F^ϕ_-(^ϕ_i)> 0 and F^ϕ_+(^ϕ_i)> 0. 


We call ^ϕ the jump latitudes of ϕ and L_par[^ϕ] the configuration of ϕ.  
If ϕ(0) = 1, we say ϕ is a unit RLD solution.  
If ϕ is extendible to  C^∞_||(∖ L_par[]) 
we say ϕ is smooth at the poles.  


If ^ϕ_i is a jump latitude of ϕ, note that <ref>.(iii) implies ∂ϕ is not defined at [_i^ϕ].  
Thus, the jump latitudes of ϕ and their number are uniquely determined by ϕ. 



Let ϕ be an RLD solution.  
We define

    ^ϕ := ( F^ϕ_1-, F^ϕ_1+, F^ϕ_2-, …, F^ϕ_k+) ∈^2k_+,
    
    ^ϕ := ( F^ϕ_i)_i=1^k  ∈^k_+,
      ^ϕ := (σ^ϕ_i)_i=1^k-1∈^k-1, 
      ^ϕ := ( ξ^ϕ_i)_i=1^k ∈^k,

where for i=1, …, k  and j=1, …, k-1,

    F^ϕ_i± := F^ϕ_± (_i),    
    F^ϕ_i := F^ϕ_+(_i)+F^ϕ_-(_i),    
    e^σ^ϕ_j = F^ϕ_j+1/F^ϕ_j,   ξ^ϕ_i = F^ϕ_i+ - F^ϕ_i-/F^ϕ_i+ + F^ϕ_i-.
 
We define ^ϕ: = (^ϕ, ^ϕ) ∈^k-1×^k and call the entries of ^ϕ the flux ratios of ϕ.   
If ^ϕ =  0 we call ϕ balanced.        
Finally we define

    F^ϕ_avg : = 1/2k|  ^ϕ|_ℓ^1 =  1/2k| ^ϕ|_ℓ^1.





Using (<ref>) (see also Remark <ref>), we recover ^ϕ from F^ϕ_1 and ^ϕ: 

    F^ϕ_1± = 1/2(1±ξ^ϕ_1)F^ϕ_1 , 
         F^ϕ_i± =  1/2(1±ξ^ϕ_i) ( e^∑_l=1^i-1σ^ϕ_l)  F^ϕ_1, 
         i>1.

In Proposition <ref> we construct RLD solutions ϕ by prescribing F^ϕ_1- and ^ϕ.




 §.§ Existence, uniqueness, and estimates

ab


If ϕ is an RLD solution, it is immediate from (<ref>) that ∂ϕ (recall <ref>) is decreasing on any domain on which it is smooth.  The scale invariant flux F^ϕ_± enjoys a similar monotonicity:


Suppose ϕ∈ C^∞_({∈ (a, b)}), ϕ> 0 and _χϕ =0.  For ∈ (a, b),  

    d F^ϕ_-/d() = 2^2  + (F^ϕ_-())^2>0.




Let t∈ (a, b) and Ω = {∈ (a, t)}.  By the divergence theorem and that _χϕ =0, 

    ∫_∂Ω⟨∇ϕ/ϕ, η⟩  = ∫_ΩΔ( logϕ) =∫_Ω( Δϕ/ϕ- |∇ϕ|^2/ϕ^2)= -∫_Ω( 2^2  +  |∇ϕ|^2/ϕ^2).

Using that η = ∂/∂ when = t and differentiating the above with respect to t yields the lemma.  



Suppose ϕ∈ C^∞_({∈ [a, b]}), ϕ>0, and _χϕ = 0. Then

    F^ϕ_-(b) + F^ϕ_+(a) = 2( tanh b - tanh a) + ∫_a^b ( F^ϕ_-())^2d.



Follows directly from integrating (<ref>) on (a, b). 

To study RLD solutions on domains {∈ (_i, _i+1)} between successive jumps, we introduce the following auxiliary functions.   


Given ∈ (0, ∞) and F> 0, we define functions

    H^+= H^+[F; ] ∈ C^∞_(),    H^-= H^-[F; ] ∈ C^∞_()
 
by requesting that they satisfy the equations _χ H^+ = 0, _χ H^- =0 with initial data 

    H^+() = 1,      F^H^+_+()=F, 
             
    H^-() = 1,      F^H^-_-()=F.




By straightforward computations (recall Lemma <ref>), 
H^+[F;]=
H^-[-F;] and 

    H^±() = ( F_+^() ∓ F) ()  () 
    + 
    ( - F_+^() ± F) ()  ().

Note also that when ≥ 0, H^+[F; ]()  = [1, F; ]() (recall <ref>).



 For any >, H^+ = H^+[F; ] satisfies 


  * ∂ F^H^+_+/∂()=(2^2 +F^2)(H^+()/H^+())^2>0.  


  * ∂ F^H^+_+/∂ F() =(H^+()/H^+())^2>0.



Applying Lemma <ref>  to H^+[F; ] (which we denote below by H for ease of notation) yields

    d F^H_+/d()  + (F^H_+())^2 = - 2 ^2 .

Differentiating (<ref>) with respect to  and switching the order of differentiation gives

    ∂/∂( ∂ F_+^H/∂)+2  (∂ F_+^H/∂)F_+^H= 0.

After multiplying through by the integrating factor H^2, this is equivalent to 

    ∂/∂(  ∂ F_+^H/∂ H^2) = 0,

from which we conclude

    ∂ F_+^H/∂() =∂ F^H_+/∂()(H()/H())^2.
  
Finally, differentiating both sides of the equation F^H^+[F; ]_+() = F with respect to  yields 

    ∂ F^H_+/∂() = - ∂ F^H_+/∂() = 2^2 + F^2,

where the last equality follows from <ref>.  This completes the proof of (i). (ii) follows in an analogous way by differentiating (<ref>) with respect to F and observing that ∂ F_+^H/∂ F() = 1.







Suppose 
a>0, 
Ω = {∈ (-a, a)}, 
∈^j_+ is as in <ref> with _j<a, 
and 
ϕ is a function on Ω which satisfies ϕ∈ C^∞_||(Ω^),  
ϕ(0) = 1, ϕ>0, and _χϕ = 0 on Ω^. 
Finally, suppose that 
F^ϕ_±(_i) = F_i± for some positive numbers F_i±, i∈{1, …, j}.  
Then

    ϕ={
    A_0+B_0  on  {∈ [-_1, _1]},
    
    
    A_1+B_1  on  {∈ [_1, _2]},
    
    …
    
    A_j + B_j  on  {∈ [_j, a)},
    .,

where the coefficients A_i, B_i, satisfy the recursive equations 

    A_0 = 1       A_i   = A_i-1- ϕ(_i)(F_i++F_i-)(_i),
    
    B_0=0       B_i   = B_i-1 + ϕ(_i)(F_i++F_i-)(_i) 
       ( 0< i ≤ k) .




Clearly A_0 =1 and B_0 = 0, since ϕ(0) =1 and ϕ extends evenly across = 0.  Now fix i∈{1, …, j}.  By Remark <ref>,  ϕ = ϕ(_i)H^-[ F_i-; _i] on {∈ [_i-1, _i]} and ϕ = ϕ(_i)H^+[ F_i+; _i] on {∈ [_i , _i+1]} and explicitly,

    ϕ(_i)H^+()    = ( F_+^(_i)- F_i+)ϕ(_i) (_i)()+( - F_+^(_i)+F_i+)ϕ(_i) (_i)()
    ϕ(_i)H^-()    = ( F_+^(_i)+ F_i-)ϕ(_i) (_i)()-(  F_+^(_i)+F_i-)ϕ(_i) (_i)().

Subtracting the second of these equations from the first yields (<ref>).





Given _1 ∈ (0, _root) and 

    = (, )  = 
    (   (σ_i )_i=1^∞  ,   (ξ_j)_j=1^∞ )∈ℓ^1(^) ⊕ℓ^∞( ^)

satisfying | |_ℓ^∞ < 1/10,   
there is a unique unit RLD solution 
= [_1; ] 
satisfying the following. 


(a). 
_1^=_1.   


(b). 
^=. |_k where 
k= k[_1; ]∈ is the number of jump latitudes of  (recall <ref>)  
and  
. |_k := (   (σ_i )_i=1^k-1  ,   (ξ_j)_j=1^k  )∈^k-1×^k. 

Moreover the following hold.


  * k[_1; ] is a nonincreasing function of _1.  
Further,
for each  as above 
there exists a decreasing sequence {a_0, := _root, a_1, , …}  
such that k[_1; ] = k if and only if _1 ∈ [a_k, , a_k-1, ). 
Moreover each a_k, depends only on 
. |_k (defined as above). 


  * _2^, …, _k^ are increasing smooth functions of _1 for fixed .  


  * [a; ] is smooth at the poles if and only if a = a_k, for some k≥ 1. 


  * The restriction of [_1; ] on any compact subset of [0, ∞)  depends continuously on _1 and .  





We first prove that there is at most one unit RLD solution  satisfying (a) and (b). 
By the symmetries, it follows that = on {∈ [-_1,
_1]}.  
Furthermore,
by Remark <ref> and Lemma <ref>,  has a unique local
extension beyond _1.  
Next, the flux monotonicity—Lemma <ref>—and the requirement 
in Definition <ref>.(i) that >0, 
inductively determine all of the jump latitudes  uniquely
using <ref>.  
From this,  is determined uniquely by <ref>.


We next construct a family [_1; ] of RLD solutions, 
parametrized by _1 or equivalently F^_1, with flux ratios .  
By Lemma <ref>, the restriction . F^_-|_(0, _root): (0, _root)→ (0, ∞) is an orientation-preserving homeomorphism.  
Because any unit RLD solution coincides with  on {∈ [0, _1]}, 
it follows there is a 1-1 correspondence between choices of 
_1 ∈ (0,  a_0, := _root) and F_1 = F_1(_1) : = 2/1- ξ_1 F^_-(_1)∈ (0, ∞).  

Let _1 ∈ (0, _root).  
By Remark <ref> and Lemma <ref>, there is a unique extension [_1; ] of 
. |_{∈ (-_1, _1)} to a maximal domain {∈ (-a, a) } such that the hypotheses of Lemma <ref> hold, 
where the coefficients F_i± (recall the notation of Lemma <ref>) 
are defined by (recall (<ref>))

    F_1± := 1/2(1±ξ_1)F_1 ,      F_i± := 1/2 (1±ξ_i) ( e^∑_l=1^i-1σ_l) F_1,    i>1.


To show that  is an RLD solution, we must show that a = ∞ and  has finitely many jump latitudes.  
By Remark <ref> and Lemma <ref>,

    2 F_1    ∼_exp(||_ℓ^1+3||_ℓ^∞)( F_i+1- + F_i+)
       ∼_exp(||_ℓ^1+3||_ℓ^∞)(2( tanh_i+1 - tanh_i) + ∫__i^_i+1( F^_-())^2d).

This implies a lower bound on _i+1 - _i which is uniform in i.  Therefore a = ∞.  We next show there are finitely many jump latitudes by estimating an upper bound for _k. 
Suppose  has a jump at _j+1. 
On {∈ (_j, _j+1)},  coincides with

    (_j) H^+[F_j+; _j] = ( F^_+(_j) - F_j+) (_j)(_j) () 
    
      +( - F^_+(_j) + F_j+) (_j)(_j) ().

Since  has a jump at _j+1, it follows that
F^_+ (_j) > F_j+.  
Since
 
    F^_+ ()= ^2/tanh↘ 0   as  →∞,
 
 and F_j+∼_exp(||_ℓ^1+3||_ℓ^∞) F^_-(_1), this implies an upper bound for _k-1 depending only on _1 and .  
 This establishes the existence of the family of RLD solutions [_1; ]. 
 
We next prove (i).  From Definition <ref> and Lemma <ref>, it
follows that
 
    f_1():=  F^_+() - 1+ξ_1/1-ξ_1 F^_-()

 is monotone on (0, _root) and moreover satisfies

    lim__1↘ 0 f_1() = ∞,   and  lim__1 ↗ a_0,  f_1(_1) = - ∞.


We then define a_1, to be the unique root of f_1 in (0,
a_0, ).  By (<ref>), a_1, depends only
on ξ_1.  There are three cases.

Case 1: _1> a_1,.  It follows from Remark
<ref>, (<ref>), and (<ref>) that A_1(_1)< 0.  Since
B_1>0 by Lemma <ref>, the monotonicity of  and 
imply that ∂ϕ>0 for all >_1.  
Hence k[_1; ] = 1.

Case 2: _1 = a_1,.  Then A_1(_1) = 0.
By Lemma <ref>, ϕ coincides with a positive multiple of
 on a maximal domain to the right of _1, which must be {∈ (_1, ∞)}  since F^_+ is strictly decreasing and
lim_→∞F^_+() = 0.  Consequently
[a_1, ; ] is smooth at the poles and
k[a_1, ; ] =1.


Case 3: _1 < a_1,.  Then A_1 >0.  Since  is monotone and lim_↗∞∂()= -∞, there exists _1max∈ (_1, ∞) with the property that ∂()>0 when ∈ (_1, _1max) and  attains a local maximum at _1max.  In particular, F^_-(_1max) = 0.  By the flux monotonicity, there exists _2 ∈ (_1max, ∞) 
such that (_1)H^+[F_1+; _1](_2) = F_2-.  
Since  remains positive, _2∈ L_par and  has its second jump latitude at _2.  

We next prove _2(_1), _3(_1),… are increasing functions of _1.  We first show this for _2(_1).  By Lemma <ref>, F^_1 is an increasing function of _1.  By <ref>, F^_2- = 1-ξ_2/2 e^σ_1 F^_1.  By combining this with both parts of Lemma <ref>, it follows that _2(_1) is strictly increasing. Using this fact and replacing _1 by _2 and _2 by _3 in the preceding argument shows that _3(_1) is strictly increasing, and inductively in the same way _j(_1) is strictly increasing for 2< j≤ k.  This proves (ii).  

By the discussion above, [a_1, ; ] is smooth at the poles and k[_1; ] = 1 for _1∈ I_1.  
By a straightforward inductive argument, 
there are unique a_2, >a_3, > …> 0, 
where a_j, depends only on the first j-1 entries of  and the first j entries of , and intervals 
[a_i, , a_i-1, ), j ∈ such that for each i≥ 1, [a_i, ; ] is smooth at the poles,  
k[a_i, ; ] = i, and k[_1; ] = i when _1∈ [a_i, , a_i-1, ).  
This concludes the proof of (i) and (iii).  
(iv) follows from the continuous dependence in of the coefficients in the conclusion of Lemma <ref> on _1 and the continuous dependence of the _2, _3, … on _1 from (ii).



One construction in <cit.> produces surfaces whose configurations consist of points 
on two parallel circles with latitudes whose absolute values are close to (in the notation of <cit.>) _balanced∈ (0, _root).  
Using (<ref>) to relate  and  coordinates, we have that _root = cos_root and (a_1, ) = cos_balanced.  



If = [_1; ] is as in Proposition <ref>.(iii), 
recall from  <ref>.(ii) that ^_i is an increasing function of _1.  
In Lemma <ref> and Corollary <ref> we estimate the derivatives precisely. 



Let = (, ), 
_1 ∈ (0, _root),  
= [_1; ], 
and = ^[ _1; ] be as in Proposition <ref>.  
Let k∈. 
 has at least k jumps if and only if 
_1 ∈ (0, a_k-1, ). 
The kth jump latitude _k depends then only on _1 and . |_k.  
_k can be considered as a smooth function defined on the domain S_k⊂×^k-1×^k where 

    S_k:=
    {  (  _1   ,
      (σ_i )_i=1^k-1  ,   (ξ_j)_j=1^k  )   :  _1 ∈ (0, a_k-1, )  and  |ξ_j|<1/10  for  j=1,…,k }.
 
Alternatively we can consider each _k as a smooth function of 
F_1=F_1^ and . |_k and then we have for k=1 (where _1 is a function of F_1 and ξ_1 only) 

    (2 ^2 _1 + ( F^_1-)^2)∂_1 /∂ F_1   = 1/2(1 - ξ_1), 
        (2 ^2 _1 + ( F^_1-)^2)∂_1 /∂ξ_1   = - 1/2 F_1,
 
and for k>1 the recursive formulas (note S_k⊂ S_k-1) 

    ( 2^2 _k + (F^_k-)^2)∂_k/∂ F_1
    = 
    ( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂ F_1( ϕ(_k-1)/ϕ(_k))^2+
    
    + ( ϕ(_k-1)/ϕ(_k))^21/2 (1+ ξ_k-1) ( e^∑_l=1^k-2σ_l)  + 
    1/2(1- ξ_k) ( e^∑_l=1^k-1σ_l) ,


    ( 2^2 _k + (F^_k-)^2)∂_k/∂ξ_j
    = 
    ( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂ξ_j( ϕ(_k-1)/ϕ(_k))^2+
    
    +( ϕ(_k-1)/ϕ(_k))^2δ_j(k-1)/2( e^∑_l=1^k-2σ_l)  F_1- 
    δ_jk/2( e^∑_l=1^k-1σ_l) F_1,


    ( 2^2 _k + (F^_k-)^2)∂_k/∂σ_j
    = 
    ( 2^2 _k-1 + (F^_k-1+)^2)∂_k-1/∂σ_j( ϕ(_k-1)/ϕ(_k))^2+
    
    +( ϕ(_k-1)/ϕ(_k))^2F_1/2(1+ξ_k-1)∂/∂σ_j( e^∑_l=1^k-2σ_l) 
    - F_1/2(1- ξ_k) ∂/∂σ_j( e^∑_l=1^k-1σ_l) .



The first statement follows directly from Proposition <ref>.  Below, we compute partial derivatives of _k with respect to F_1, _k-1, and the entries of .|_k, from which the smoothness claimed follows immediately.  To this end, we recall from (<ref>) and <ref>, that

    ϕ = ϕ(_k-1) H^+[ 1/2(1+ ξ_k-1)  ( e^∑_l=1^k-2σ_l)  F_1; _k-1]   on  {∈ [_k-1, _k]}.
 
Denoting H^+ as in (<ref>) by H for simplicity and using (<ref>), we find

    F^H_-(_k) = 1/2(1-ξ_k)  ( e^∑_l=1^k-1σ_l)  F_1.

Items (<ref>)-(<ref>) then follow by using the chain rule to differentiate (<ref>) and Lemma <ref> and both parts of <ref> to calculate the partial derivatives of F^H.




There is a constant _1>0 such that for all k∈, k>1 and all = (, ) ∈^k×^k+1 satisfying ||_ℓ^1+ ||_ℓ^∞< _1 and _1 ∈ (a_k+1, , a_k, ), = [_1; ] satisfies the following.


  * F^_ < C/k.

  * 1- () : C^0( {∈ [-_k, _k]})  < C/klog k and | log(_i)/(_i-1)| < C/k for 2≤ i ≤ k. 


  

Using the monotonicity of the flux in lemma <ref> we conclude that the maximum of |F^_-| is achieved at the jump latitudes. 
Using also <ref> we conclude 

    max_∈[0,∞) |F^_-()| = 
    max_1≤ i≤ k+1F^_i±∼_exp(||_ℓ^1+3||_ℓ^∞) F^_avg∼_exp(||_ℓ^1+3||_ℓ^∞)min_1≤ i≤ k+1 F^_i±.

By <ref> and the fact that  has k+1 jumps, it follows that F^_+(_k) = ^2 _k/tanh_k > F^_k+.  From this and <ref> we estimate

    F^_avg < C ^2 _k.

By using Lemma <ref> on the intervals {∈ [ 0, _1]}, …{∈ [_k-1, _k]} and summing, we find

    F^_1-+ F^_1++ ⋯ + F^_k- = 2tanh_k + ∫_0^_k( F^_-())^2   d.

Next using <ref> and (<ref>) and (<ref>) to estimate (<ref>), we find

    (2k-1)F^_ < C(1+ _k ^2 _k)< C,

from which we conclude (i).  For (ii), we observe from <ref> that on any domain on which  is smooth, F^_+() = ∂( log).  Integrating from _i-1 to _i, using the easy to see fact that for 2≤ i ≤ k, 0<_i-_i-1 <C, and estimating using part (i), 

    | log( (_i)/(_i-1))| ≤∫__i-1^_i|F^_+()| d≤C/k∫__i-1^_i d≤C/k.





Let = (, ), _1 ∈ (0, _root), = [_1; ], and = ^[_1;] be as in <ref> and suppose moreover that ||_ℓ^∞<_1/k and that  (_1, . |_k+1) ∈ S_k+1 (recall <ref>).  The following estimates hold.


  * ( 2^2 _k + (F^_k-)^2)∂_k/∂ F_1∼_C k.

  * | ( 2^2 _k + (F^_k-)^2)∂_k/∂σ_i | < C, i=1, …, k-1.

  * |( 2^2 _k + (F^_k-)^2)∂_k/∂ξ_j| < C/k, j=1, …, k.



We first prove (i).  To simplify notation in this proof, we denote

    P_k    := ( 2 ^2 _k + ( F^_k-)^2) ∂_k/∂ F_1, 
      
    Q_k-1 := ( (_k-1)/(_k))^2, 
      
    R_k :=  2 ^2 _k + ( F^_k+)^2/ 2 ^2 _k + ( F^_k-)^2,
    
    T_k-1   :=  1/2 Q_k-1 (1+ ξ_k-1) ( e^∑_l=1^k-2σ_l)  + 
    1/2(1- ξ_k) ( e^∑_l=1^k-1σ_l).

In this notation, (<ref>) from Lemma <ref> is equivalent to the equation

    P_k =  R_k-1 Q_k-1P_k-1 + T_k-1

from which we conclude by applying (<ref>) recursively

    P_k = P_1 ∏_i=1^k-1 Q_i R_i + ∑_i=1^k-1( T_i ∏_j=i+1^k-1 Q_j R_j).

From (<ref>) it follows that P_1 = 1/2(1 - ξ_1).  By <ref>, the assumption ||_ℓ^∞<_1/k, and <ref>, the following estimates hold:

    Q_i ∼_1+ C/k 1,    R_i ∼_1+C/k,    T_i ∼_C 1,      i=1, …, k-1.

Combining (<ref>) with (<ref>) completes the proof of (i).  Proofs of (ii) and (iii) are similar and use respectively (<ref>) and (<ref>) in place of (<ref>), so we omit the details. 



We are particularly interested in RLD solutions which are smooth at the poles. 
Recall from <ref> that 
[a_k,;  ] 
is the unique smooth at the poles unit RLD solution  
with k jump latitudes and flux ratios .  
Moreover by <ref>.(i), 
[a_k,;  ] 
depends only on 
. |_k. 
This motivates the following. 



Given k, , and 
. |_k as in <ref>, 
we define 

    [: k]:= 
    [. |_k : k]:= 
    [a_k,;  ],  
           = [: k ] := 
    [. |_k : k ] := 
    ^[: k]   .





Fix  as in Proposition <ref> and let 
= [  : k] be as in <ref>. 
Then 


  * For fixed i, _i = _i[ :k ] is a decreasing function of k.

  * If =, and i is fixed, _k-i = _k-i[ : k] is an increasing function of k. 



(i) follows from parts (i) and (ii) of Proposition <ref>.  (ii) follows from a dual version of Proposition <ref> where one builds [ : k] “from infinity" by taking the final flux F^_+(_k) as initial data rather than the first flux F^_-(_1) as in the proof of <ref>.  We omit the details of the argument because we do not use part (ii) in the remainder of the paper. 













 



We next establish estimates for smooth at the poles RLD solutions with small flux ratios.  Below, if we write f = O(w) for a number f and some w>0, we mean |f| ≤ C w, for some constant C independent of f and w. 



There is a constant _1>0 such that for all k∈ and all = (, )∈^k-1×^k  with ||_ℓ^1+ ||_ℓ^∞ < _1,
=[: k] satisfies the following.


  * F^_avg = 1/k +O(1/k^3) and ^2 _k ∼_C 1/k.  

  * 
	
  * F^_1-=2tanh_1 +O(1/k^3).
	
  * F^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) +O(1/k^2(k-i+1)) for i=2, …, k.
	
  * F^_k+=2(1- tanh_k) + O(1/k^2).
	

  * 1-(): C^0() ≤C/klog k and  |log(_i)/(_i-1)| ≤C/k for 2≤ i ≤ k.    If =, then also 1>(_1)> ⋯ >(_k).  

  * There is a constant C>0 depending only on _1 such that for any 1≤ i<j≤ k, 

    _j - _i    >  (1+C/k) j-i/2k  and  ^2 _i - ^2 _j ≤C(j-i)/k.






The estimates in Proposition <ref> are substantive only when k is large.  When k is small, by taking _1 small enough, using Proposition <ref>, and smooth dependence of ODE solutions on initial conditions, we maintain coarse control over the family [: k].   
More precisely we have the following alternatives for small k:  given > 0 and taking _1 small enough in terms of , there exists a constant c = c()> 0 such that for all k∈, k <, (i). 0< 1/c<  < c.(ii). 1/c< _1 and for i∈{2, …, k}, _i - _i-1∼_c 1. (iii). For i∈{1, …, k},  F^_i±∼_c 1.




Since  is smooth at the poles,

    F^_k+= F^_+(_k) = ^2 _k/tanh_k.

Using this with (<ref>) and estimating a uniform lower bound for tanh_k, we conclude 

    ^2 _k ∼_Cexp (||_ℓ^1+||_ℓ^∞) F^_avg.

By fixing _1 sufficiently small, we can ensure the constant C in (<ref>) is less than 10 and also the constants in the estimates in the statements (i)-(iii) are independent of k.  

Using Lemma <ref>, we have (where i=2, …, k)

    F^_1-   =2tanh_1 +∫_0^_1(F^_-())^2d
    
     F^_i-+F^_i-1+   = 2(tanh_i - tanh_i-1) + ∫__i-1^_i(F^_-())^2d, 
     
     F^_k+   =2(1-tanh_k) + ∫__k^∞(F^_-())^2d.

Note ∫__i-1^_i^2 /^2 d = ∫__i-1^_i1/^2  d( tanh)< C(tanh_i - tanh_i-1)/F^_ and also that
∫__k^∞( F^_-())^2 d = ∫__k^∞^4 /tanh^2  d.
Using (<ref>) and (<ref>) we estimate the integrals to obtain 
(where i=2, …, k)

    F^_1-   =2tanh_1 +_1 O((F^_avg)^2)
    
     F^_i-+F^_i-1+   =2(tanh_i - tanh_i-1) +(tanh_i - tanh_i-1)O(F^_avg), 
     
     F^_k+   =2(1-tanh_k) + O((F^_avg)^2).

By summing the estimates in (<ref>) and dividing through by k,  we find that F^_avg = 1/k - 1/kO(F^_avg),
which implies that F^_avg = 1/k+O(1/k^2).  Substituting this estimate for F^_avg into (<ref>), we find 

    2tanh_1 ∼_C 1/k,   tanh_i - tanh_i-1∼_C1/k,    
    i=2, …, k,

hence

    tanh_i ∼_C i/k,    i=1, …, k.

Summing the estimates in (<ref>) with these improved bounds yields

    kF^_avg = 1+O(1/k^2),
 
and the first part of (i) follows.  The second statement in (i) follows from the first and (<ref>).
Substituting (i) into the first and last parts of (<ref>) yields (ii).(a) and (ii).(c).  For (ii).(b), we substitute (i) into the error term of the second part of (<ref>) to get

    F^_i-+F^_i-1+   = 2(tanh_i - tanh_i-1) + 1/^2 _iO( 1/k^3).

When i ≤ k/2, (ii).(b) follows immediately from (<ref>) because (<ref>) implies ^2 _i is bounded from below by a constant independent of k.  On the other hand, when j≥ k/2, (<ref>) implies tanh_j ≥ C, so from this and (<ref>),

    ^2 _j-1 - ^2 _j     = (tanh_j - tanh_j-1)(tanh_j +tanh_j-1)
    ≥C/k.

Using  (<ref>) and summing (<ref>) and, we find for i≥ k/2

    ^2 _i ≥C(k-i+1)/k.

Substituting this bound in (<ref>) completes the proof of (ii).(b).  

We now prove (iii).  From Definition <ref>, on any domain on which  is smooth,

    F^_+() = ∂( log).

Integrating (<ref>) on intervals where  is smooth and adding, we find that for any ∈ [0, _k],

    |log() |    ≤∫_0^| F^_+()|  d≤C/k∫_0^_k  d≤C/klog k,

where we have used (<ref>) in combination with part (i) above to estimate _k = O(log k).  Since  is smooth at the poles, it coincides with a multiple of = tanh on {∈ (_k, ∞)} and so in particular the estimate in the first part of (iii) holds on {∈ (_k, ∞)}. 

 For the second statement, it is easy to see that for 2≤ i ≤ k, 0<_i-_i-1 <C.  Then estimating in the same way as above, we get

    | log( (_i)/(_i-1))| ≤∫__i-1^_i|F^_+()| d≤C/k∫__i-1^_i d≤C/k.

Next, suppose that =.  Fix i∈{ 1, …, k-1}. Integrating (<ref>) from _i to _i+1, we find that 

    log((_i+1)/(_i))    = - ∫__i^_i+1 F^_-()d.

Since  is balanced, F^_i+1- = F^_i+:= F_0 and the restriction of F^_- to [_i, _i+1] satisfies

    .F^_-|_[_i, _i+1]: [_i, _i+1]→ [-F_0, F_0]  and  dF^_-/d = 2 ^2 + F^_-()^2>0.

In particular, F^_-|_[_i, _i+1] is invertible.  Reparametrizing the integral in (<ref>) by (F^_-|_[_i, _i+1])^-1, we have

    log((_i+1)/(_i))    = - ∫_-F_0^F_0  Fd/dF dF
       = -∫_-F_0^0 F/2^2 +F^2dF-∫_0^F_0 F/2^2 +F^2dF
       <0,

since the first integral term is positive, the second is negative, and ^2 is monotonic in F by Lemma <ref>.  This completes the proof of (iii).
For (iv), assume 1≤ i<j≤ k.  By summing instances of (ii).(b) above, we find

    F^_i++⋯ +F^_j- = 2(tanh_j - tanh_i) + O( j-i/k^2(k-i+2)).

Using (<ref>), we estimate

    (j-i) exp(-||_ℓ^1-3||_ℓ^∞)F^_≤  (tanh_j - tanh_i) + O( j-i/k^2(k-i+2)).

Applying the mean value theorem to the right hand side of the preceding gives

    (j-i) exp(-||_ℓ^1-3||_ℓ^∞)F^_ < _j - _i + O( j-i/k^2(k-i+2))

and the first statement in (iv) follows from combining (<ref>) with (i) above and taking ||_ℓ^1 and ||_ℓ^∞. small enough.  The second follows from combining (<ref>) with parts (i) and (ii) above.





Proposition <ref> implies that the latitudes of L_par of RLD solutions which are close to being balanced arrange themselves in a regular way.  Indeed, recalling from (<ref>) that tanh = sin, Proposition <ref>.(i) and (ii) together imply that when =,

    sin_1    = 1/2k + O( 1/k^3),
    sin_i - sin_i-1   = 1/k + O( 1/k^2(k-i+1)),    i=2, …, k-1, 
    
    1 - sin_k    = 1/2k + O(1/k^2).

Together with elementary geometric facts about spheres, this means that for large k


  * The (extrinsic ^3) distance between planes corresponding to adjacent circles in L_par is approximately 1/k. 

  * For i=2, …, k, the area of the annulus {∈ (_i-1, _i) }⊂ is approximately 2π/k. 

Since the LD solutions we use to construct initial surfaces have configurations arising 
(in a way made precise later in Lemma <ref>) 
from RLD solutions which are approximately balanced, 
the circles of L_par in the minimal surfaces we ultimately construct are also approximately equally spaced. 






 Let k∈ and suppose that = (, )∈^k-1×^k , '= (', ') ∈^k-1×^k satisfy ||_ℓ^1+ ||_ℓ^∞< _1/k and |'|_ℓ^1+ |'|_ℓ^∞< _1/k.  Let = [ : k] and ' = [ ': k].  There is a constant C>0 independent of k such that: 
 

  * | ^' - ^|_ℓ^∞≤C/k(  | ' - |_ℓ^1+ |' - |_ℓ^∞). 


  * max_1≤ i ≤ k| tanh'_i - tanh_i | ≤C/k(  | ' - |_ℓ^1+ |' - |_ℓ^∞). 




Observe that (i) follows from the estimate

    | F^'_1 - F^_1 | ≤C/k(  |' -  |_ℓ^1 + |' - |_ℓ^∞),

because for any i∈{1, …, k} we may estimate (assuming (<ref>) for a moment)

    2| F^'_i+ - F^_i+|    = 
    | (1+ ξ'_i ) ( e^∑_l=1^i-1σ'_l)  F^'_1 - (1+ ξ_i)  ( e^∑_l=1^i-1σ_l)  F^_1| 
       ≤|  (1 + ξ'_i)  ( e^∑_l=1^i-1σ'_l)  -  (1 +ξ_i)  ( e^∑_l=1^i-1σ_l) | F^'_1
    +   (1 + ξ_i)  ( e^∑_l=1^i-1σ_l) | F^'_1 - F^_1| 
       ≤C/k(  |' -  |_ℓ^1 + |' - |_ℓ^∞)

and a corresponding bound for | F^'_i- - F^_i-| follows in an analogous way.  We now prove (<ref>).

Fix k∈, let _1 be as in <ref>, and 
consider the map 
defined by

    ( F_1,  ) = F^[ _1; ]_+(_k) - F^_+(_k),

where [ _1; ] is as in <ref> and _1 is chosen so F^_1 = F_1.  Clearly, ( F_1, ) = 0 if and only if  is smooth at the poles.  Now let ∈^-1({0}) be arbitrary.

It follows from Lemma <ref> and <ref> that  is C^1; below we estimate the partial derivatives of  at .

Differentiating (<ref>) with respect to F_1 and using (<ref>) and <ref>, we compute

    . ∂/∂ F_1 |_   = 1/2(1+ ξ_k) ( e^∑_l=1^k-1σ_l)  + (2 ^2 _k + (F^_k+)^2) .∂_k/∂ F_1|_.

By combining (<ref>) with Corollary <ref>, we estimate that for j∈{1, …, k} and i∈{1, …, k-1},



    .∂/∂ F_1|_∼_C k,     | .∂/∂σ_i|_| ≤ C,     |. ∂/∂ξ_j|_| ≤C/k,

where C>0 is a constant independent of k.

By the implicit function theorem, locally around (F_1, ) ∈^-1( {0}), ^-1({ 0 }) is a graph over  and moreover (abusing notation slightly), for i∈{1, …, k-1} and j∈{1, …, k}

    .∂ F_1/∂σ_i|_ = - (. ∂/∂ F_1|_)^-1.∂/∂σ_i|_,     .∂ F_1/∂ξ_j|_= - ( .∂/∂ F_1|_)^-1. ∂/∂ξ_j|_.

(i) follows by combining this with the estimates (<ref>).  The proof of (ii) is similar to the proof of (i), but we omit it since we will not use (ii) in the remainder of the paper. 












§ LINEARIZED DOUBLING (LD) SOLUTIONS













 §.§ LD solutions

ab
 

Given a finite set L⊂ and a function τ:L→,
we define a linearized doubling (LD) solution of configuration (L,τ) to be a function 
φ∈ C^∞(∖ L) which satisfies the following conditions, 
where τ_p denotes the value of τ at p:


  * φ=0 on ∖ L. 

  * ∀ p∈ L there is a smooth extension across {p}, 
_p∈ C^∞( {p}∪(∖ (L∪{-p}) ) ),   
such that 

    _p=φ-τ_p   G^^2∘^g_p    on  ∖ (L∪{-p}),
 
where G^^2∘^g_p   
is as in <ref>. 





By Lemma <ref>, 
the class of LD solutions  
and each corresponding τ_p, 
do not depend on which Green's function we use in Definition <ref>.(ii) above. 
Note however that  and its domain do. 




Given L and τ as in Definition <ref>, 
and also a finite dimensional space [L] ⊂ C^∞(), 
we define a linearized doubling (LD) solution of configuration (L,τ,w) to be a function 
φ∈ C^∞(∖ L) which satisfies the same conditions as in <ref>, 
except that condition (i) is replaced by the following:

 
abc 
(i'). φ=w∈[L] ⊂ C^∞() on ∖ L.

 

Note that LD solutions in the sense of Definition <ref> are LD solutions with w=0 in the sense of <ref>.  
Existence and uniqueness of -symmetric LD solutions modulo [L] is a consequence of the symmetries of the construction: 

  

We assume given -invariant L,τ,w, 
where L= L[;m] is as in <ref>,  
τ:L→, and w∈[L].  
There is then a unique -invariant LD solution modulo [L] 
of configuration (L, τ, w),  
φ = φ[L, τ, w]. 
Moreover, the following hold.


  * φ and _p each depend linearly on (τ, w).


  * φ_∈ C^0(∖(L∩{p_N,p_S}) )
and φ_ is smooth on 
()^ 
where it satisfies the ODE φ_=0.


If w=0, then we also write φ = φ[L; τ] and 
φ is the unique -invariant LD solution of configuration (L, τ) as in <ref>.


This is Lemma 3.10 of <cit.>.










 §.§ Estimates for LD solutions on the cylinder

ab


The next lemma asserts a balancing law for LD solutions φ which restricts their averages φ_avg. 



Suppose φ = φ[L[; m]; τ] is as in 
<ref>, where L[; m] and τ are as in <ref>.  
Then 

    m τ_i = φ_avg(_i) F^φ__i,    i=1, …, k.




The proof is similar to the proof of <cit.>.  
Fix i ∈{1, …, k}.  
For 0<ϵ_1<<ϵ_2 we
consider the domain 
Ω_ϵ_1,ϵ_2:=
D^χ_L_par[_i](ϵ_2) 
∖ 
D^χ_L_i(ϵ_1).
By integrating _χφ=0 on Ω_ϵ_1,ϵ_2
and integrating by parts we obtain

    ∫_∂Ω_ϵ_1,ϵ_2∂/∂ηφ  +   2 ∫_Ω_ϵ_1,ϵ_2( ^2 ) φ = 0,

where η is the unit outward conormal vector field along ∂Ω_ϵ_1,ϵ_2.  By taking the limit as ϵ_1→ 0 first
and then as ϵ_2→ 0 we obtain using the logarithmic
behavior near L that 

    m τ_i = ∂_+ φ_avg(_i) + ∂_-φ_avg(_i).

 (<ref>) follows from  (<ref>) after appealing to the definition of 
F^φ_avg_± (recall <ref> and <ref>.(iii)).




We assume from now on k, m∈ are fixed and m satisfies m >
where = (k)>0 is a constant which can be taken as large as necessary in terms of k.
  

In the next lemma we convert RLD solutions  to LD solutions Φ whose non-oscillatory part 
in the sense of <ref> is a multiple of . 
The overall scale of the LD solutions used is determined later in <ref>,  
while at the moment we only fix their scale by a suitable normalizing condition. 
The purpose of the rest of this section is to understand and estimate carefully 
Φ by decomposing it to well described parts  and , 
and an error term  which is estimated in Proposition <ref>. 



Given = (, )∈^k-1×^k as in <ref> 
and m as in <ref>, 
there is a unique -invariant LD solution (recall  <ref>)

    Φ = Φ [ : k,m] := φ[L; τ'],

characterized by the requirements that
 
 
  * ϕ = ϕ[: k,m]:=Φ_ is a multiple of [ : k]
 
  * L = L[  [ : k ]  ; m   ] (recall <ref>) 
 
  * τ'_1 = 1 (normalizing condition).
 
 
Moreover,  the following hold.
 

  * For i∈{1, …, k}  we have 
τ_i' = ϕ(_i)/mF^ϕ_i. 
Moreover τ_i' is independent of m and satisfies 
τ_i' = τ_i' [ : k] := 
[: k](_i)/[: k ](_1)( e^∑_l=1^i-1σ_l).

  * ϕ[: k,m]   =  m/ [:k](_1)    F^[:k]_1  [: k ].

  * On Ω[_i;m], ϕ = _i + _i, where

    _i    :=  
    [ ϕ(_i), ; _i] =  ϕ(_i) [1, 1/2(F^ϕ_+(_i) - F^ϕ_-(_i)) ; _i],
    _i    :=   [τ'_i/2; _i].





Suppose Φ is as in (<ref>) and satisfies (a)-(c).  
Let c be such that ϕ=c and  i∈{1, …, k}. 
Using Lemma <ref> to solve for τ'_i, we immediately conclude τ'_i = ϕ(_i) F^ϕ_i /m; 
furthermore using Lemma <ref>, (a)-(c) above, and (<ref>), we compute
 
    τ'_i    = τ'_i/τ'_1 = ϕ(_i)/ϕ(_1)F^ϕ_i/ F^ϕ_1 = (_i)/(_1) F^_i/F^_1 = (_i)/(_1)( e^∑_l=1^i-1σ_l) ,
    
    1    = τ'_1 = ϕ(_1)/m F^ϕ_1 =  c (_1)/m F^_1.

We conclude from these equations that (a)-(c) imply (i) and (ii).  
In particular, the second equation in (i) determines τ' and hence uniqueness follows from Lemma <ref>.   

To prove existence we define L by (b) and τ' by the second equation in (i). 
We then define Φ by (<ref>) and we verify that Φ_ = c, where c is defined by c (_1)  F^_1 = m: 
Let i∈{1, …, k}. By Lemma <ref>, it follows that m τ'_i = Φ_(_i) F^Φ__i.  
By the definitions of τ'_i and c, we have m τ'_i = c (_i ) F^_i.  
Since F^_i = F^c_i, by equating the right hand sides of the preceding equations, 
we conclude that the function f := c - Φ_avg satisfies 

    ∂_+f(_i)+∂_-f(_i) = 0,    i=1, …, k.

This amounts to the vanishing of the derivative jumps of f at each _i.  
Clearly f is smooth at the poles and satisfies _χ f = 0 in between the _i.  
Hence we conclude f∈ C^∞() and satisfies f = 0 everywhere.  
By the symmetries of f, we conclude f = 0.

It remains to check (iii).  
The vertical balancing equation (<ref>) implies

    m τ'_i =  ∂_+ ϕ(_i) + ∂_-ϕ(_i),

so from the definition of  in <ref>,

    ∂_+_i (_i)  = ∂_-_i (_i)= ∂_+ϕ+∂_-ϕ/2(_i).
 
Therefore, ϕ - _i satisfies

    ∂_+ (ϕ - _i)(_i) = ∂_+ ϕ - ∂_- ϕ/2(_i) = - ∂_-(ϕ - _i)(_i).

Hence, ϕ - _i∈  C^1_(Ω[_i;m]) and _χ(ϕ-_i)  = 0.  
By uniqueness of ODE solutions,  ϕ - _i = _i. 
The second expression in (<ref>) follows from this by <ref>.















For i=1, …, k we define numbers A_i: = τ'_i logδ (recall (<ref>)).  
We define ∈ C^∞_(∖ L) by requesting that 
it is supported on D^χ_L(3δ) where it is defined by 

    = 
      [2δ, 3δ; ^χ_p_i](τ'_i G_p_i - [A_i,0; _i],  0)
          on     D^χ_p_i(3δ),
 
for i=1, …, k, 
where the discs D^χ_p_i(3δ) are disjoint by 
Assumption <ref> and Proposition <ref>. 




The following hold.


  * : C_^j(  ∖ D^χ_L(δ) ,  )≤ C(j) and  is supported on D^χ_L(3δ)∖ L.


  * : C_^j(   D^χ_L(3δ)∖ D^χ_L(2δ)   ,   ) ≤ C(j)/m.



The statement in (i) on the support follows from Definition <ref> and the corresponding estimate follows from Lemma <ref> and Definition <ref> (observe the scale is such that the cutoff  has all derivatives bounded in the  metric).  For (ii), because  is supported on D^χ_L(3δ)∖ L, it will suffice to prove the estimate 

    : C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ),   )≤ C(j)/m.

Recalling Definition <ref>, we have on D^χ_p_i(3δ)∖ D^χ_p_i(2δ)

    =  [2δ, 3δ; ^χ_p_i](τ'_i   G_p_i, 0) - 
    	 [2δ, 3δ; ^χ_p_i](  [A_i, 0; _i], 0)
       := (I)+(II).

From Lemma <ref>.(iii) and the uniform bounds on the cutoff , we have (I):  C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ)   ,   ) ≤ C(j)/m.  By Lemma <ref>.(iii), (II):  C^j( D^χ_p_i(3δ)∖ D^χ_p_i(2δ)   ,   ) ≤ C(j) (log m) m^-3, and these estimates complete the proof of (ii).



By Definition <ref> and Lemma <ref>, 
we have for each p_i ∈ L that = τ'_iG_p_i- [A_i, 0; _i] on D^χ_p_i(2δ).  
Using this,  Definition <ref>.(ii) and (<ref>), we see that Φ- can be extended smoothly across L.

In the next subsection, we will estimate Φ.  
The key tool is that Φ is well approximated by ϕ away from L.  
ϕ is well understood by the analysis in Section <ref>, in particular by <ref>, <ref> and <ref>.  
Lemma <ref> motivates the definition of a smooth rotationally invariant function , 
which is a dominant term in the decomposition of Φ in <ref>.
Note that Assumption <ref> and Proposition <ref> imply that Ω[_i;m] ∩Ω[_j;  m] = ∅ when i≠ j.




We define ∈ C^∞_||( ) by requesting that
 
    := 
    [ 2/m, 3/m; ^χ_[_i]] ( _i, ϕ) = ϕ - [ 2/m, 3/m; ^χ_[_i]] (_i, 0)

 on Ω[_i;m], i=1, …, k (recall (<ref>), (<ref>))
and that = ϕ on ∖ D^χ_L_par(3/m).




We are now ready to define a decomposition Φ =  +  +.  
The third term  we treat as an error term to be estimated (see Proposition <ref>).


We define
Φ', E'∈ C^∞_() by requesting that on
∖ L 
(recall <ref> and <ref>),  

    Φ=++,      E'=-_( + ).
 





The decomposition in Definition <ref> in some way combines the decomposition Φ =  + Φ”  <cit.> with the semilocal decomposition Φ” =  + [ϕ_1 - A_1, (_1); _1] <cit.>.  There are the following differences:


  * Our definition of  transits to zero away from L, whereas the  in <cit.> transits to A_1.
 
  * Here,  is defined globally, whereas in <cit.>,  is defined only on Ω_1[_1;m].
 








 §.§ Estimating  and 

ab


We estimate the average and oscillatory parts of Φ separately.  Taking averages of both sides of the equation Φ =+ + (recall Definition <ref>) we have _avg, _osc, E'_avg, E'_osc∈ C^∞_sym() and 

    = _avg + _osc,    E'=E'_avg+E'_osc  on  .

Furthermore, since _χΦ vanishes by Definition <ref> and _χ is rotationally covariant,

    _ = E',
        __ = E'_,
        __ = E'_  on  .


Because _χϕ =_χ = 0, it follows that E'_ = __ is supported on 
 (D^χ_( 3/m) ∖ D^χ_( 2/m)) ⋃ D^χ_( 3δ).
We have the following characterization of _avg.


_ is supported on D^χ_(3/m).  On Ω[_i;m], i=1, …, k,

    _avg = [ 2/m, 3/m; ^χ_[_i]] ( _i, 0 ) ,   on  Ω[_i;m]∖Ω'[_i;m]
    _i -_,   on  Ω'[_i;m].



By taking averages of the equation Φ =  +  + and rearranging, we find

    _ = ϕ -  - _.

Equation (<ref>) follows from this decomposition after substituting the expression for  from <ref> and recalling that = 0 on Ω[_i;m]∖Ω'[_i;m]. 


The decomposition Φ = +  + is designed so that  is small in comparison to  (cf. Proposition <ref> below).  To estimate , we will use that it satisfies the equation _ = E'.  First we establish relevant estimates for E', E'_avg and E'_osc.  


E' vanishes on D^χ_L(2δ) and E'_ is supported on D^χ_(3δ). For each i∈{1, …, k}, the following hold.


  * 

  * E':C_^j( Ω[_i;m] ,  )≤ C(j)

  * E'_:C_^j(Ω[_i;m]  ,  )≤ C(j)

  * E'_:C_^j( Ω[_i;m],  )≤ C(j)


  * 

  * E' :C_^j(  D^χ_[_i](3δ) ,   )≤ C(j)/m

  * E'_ :C_^j(   D^χ_[_i](3δ)  ,   )≤ C(j)/m 

  * E'_ :C_^j(   D^χ_[_i](3δ)  ,   )≤ C(j)/m.





The statements on the support of E' and E'_ follow from <ref>, <ref>, and <ref>.
Next, note that parts (b) and (c) of items (i) and (ii) follow from part (a) of the respective items by taking  averages and subtracting, so it will suffice to prove part (a) of (i) and (iii).
It follows from <ref> that on Ω[_i; m], 

    E' = - _(  + ).

On Ω[_i; m]∖Ω'[_i;m], it follows from this, <ref> and <ref> that
 E' = _[ 2/m, 3/m; ^χ_[_i]] ( _i, 0 ).  Thus, when restricted to Ω[_i; m]∖Ω'[_i;m], the bound in (i).(a)  follows from <ref> and the uniform bounds of the cutoff.  By <ref> and <ref>, E' vanishes on Ω[_i;m]∖ D^χ_[_i](3δ).
On D^χ_[_i](3δ), note that _=0. Since _ = 0 on  D^χ_L(2δ), when restricted to D^χ_[_i](3δ), the required bound in (i).(a) follows from Lemma <ref>.(i).  

For (ii).(a), consider D^χ_[_i](3δ).  Applying  to both sides of (<ref>) yields that E' = - _ on D^χ_[_i](3δ).  Since E' vanishes on D^χ_[_i](2δ), it is only necessary to prove the estimate on D^χ_[_i](3δ)∖ D^χ_[_i](2δ).  Using <ref>.(ii) to switch the order of _ and , we find

    E' = - _   - 2m^-2 [ ^2 ]    E'   on   D^χ_[_i](3 δ) ∖ D^χ_[_i](2δ).

The estimate in (ii).(a) follows from this equation after using Lemma <ref>.(ii) above to estimate the first term and Lemma <ref>.(iii) and (i).(a) above to estimate the second. 






Given 
E∈ C^0,β_() with E_≡ 0 and E supported on D^χ_[_i](3δ) for some i∈{1, …, k}, there is a unique u ∈ C^2,β_() solving _ u = E and satisfying the following.


  * u_=0.

  * u: C^2, β_sym(, , e^-m| || - _i | )≤ C
  E: C^0, β_sym(D^χ_[_i](3δ),  ). 

  * u:C^2,β_(D^χ_[_i](3δ)  ,  )≤
 Cm^-3 E : C_^0,β(D^χ_[_i](3δ),   )
+C E: C_^0,β(D^χ_[_i](3δ)  ,  ).

















Since the kernel of _ : C^2,β_()→ C^0,β_() is trivial by the symmetries (recall Lemma <ref>), the existence and uniqueness of u is clear.  Since _ is rotationally covariant, _ u_avg = E_avg = 0 and (i) follows. 

The equation _ u = E is equivalent to 

    ( Δ_χ  + 2^2 ) u =  m^2E.

We use separation of variables to establish the estimate (ii) (see <cit.> 
for a similar technique). Recall {1, (cos n θ)_n∈,  (sin n θ)_n∈} is a basis for L^2(^1).  Define L^2-projections

    E_0()    = 1/2π∫_0^2π E(, θ) dθ
    
    E_n, even()    = 2/π∫_0^2π E(, θ) cos (nθ) dθ
    
    E_n, odd()    = 2/π∫_0^2π E(, θ) sin (nθ) dθ.

The assumption E_ = 0 implies E_0 ≡ 0.  By the symmetries, E_n, odd≡ 0 and E_n, even≡ 0 for all n such that m does not divide n.  The Fourier expansion of u then satisfies 

    u(, θ) = ∑_q=1^∞u_mq, even() cos(mqθ)+ 
    ∑_q=1^∞u_mq, odd() sin(mqθ)

for appropriate functions u_n, even(), u_n, odd() described as follows.  Separating variables leads us to consider the eigenspace {  u ∈ C^∞_ (): _χ u = n^2 u},
which is spanned by

    u_± n = (± n - tanh) e^± n.

From this, it can be checked that for any n≥ 2, _χ - n^2 has associated Green's function

    _n(, ξ) = 
    1/2n(1-n^2)
    e^n( - ξ) (n+tanhξ)(n - tanh)    for  ≤ξ
    
    e^n(ξ - )(n - tanhξ)(n+ tanh)   for  ≥ξ.

For each q∈, we have then using that E is supported on Ω[_i;m]

    u_mq, even()    =m^2 ∫_-∞^∞_mq(, ξ) E_mq, even(ξ) dξ
       = 2m^2/π∫__i - 3δ^_i+3 δ∫_0^2π_mq(, ξ) E(ξ, θ) cos(mq θ) dθ dξ ,
    u_mq, odd()    = 2m^2/π∫__i - 3δ^_i+3 δ∫_0^2π_mq(, ξ) E(ξ, θ) sin(mq θ) dθ dξ.




A straightforward estimate of (<ref>) implies that for all q∈,

    | _mq, even| ≤C/q^2 E: C^0_sym(D^χ_[_i](3δ),   ),
    | _mq, odd| ≤C/q^2 E: C^0_sym(D^χ_[_i](3δ),   ).
 
Therefore, 

    u: C^0_sym(D^χ_[_i](3δ),  )≤ C E:  C^0_sym(Ω[_i;m],  ).
 
Since _ u = E, the C^0 bound above implies with the Schauder estimates that

    u: C^2, β_sym(D^χ_[_i](3δ), ) ≤ CE:  C^0, β_sym(D^χ_[_i](3δ),  ).
 
Combining (<ref>) with the exponential decay from (<ref>) yields

    u: C^2, β_sym(, , e^-m| || - _i |)≤ C   E: C^0, β_sym(D^χ_[_i](3δ),  ).

This proves (ii). 
Applying  to both sides of (<ref>) and using Lemma <ref>.(iii), we obtain

    _ [   u ]=- 2m^-2 [ ^2  ]   u +   E.

Although E-  2m^-2  [ ^2  ]   u is not supported on D^χ_[_i](3δ), it has average zero, so a straightforward modification of the argument leading up to (<ref>) by replacing the assumption that the inhomogeneous term is compactly supported  with the assumption (from (ii) above) that the right hand side has exponential decay away from D^χ_[_i](3δ), we conclude that

    u: C^2, β_(D^χ_[_i](3δ) ,  )   ≤ C   E: C^0, β_(D^χ_[_i](3δ) ,  )
       + Cm^-2  [ ^2  ]  u: C^0, β_(D^χ_[_i](3δ) ,  ).

(iii) follows after using (<ref>), Lemma <ref>.(iii), and part (i) above to estimate the last term.
 

Recall that E'_ is supported on D^χ_L_par(3δ).  With this and Lemma <ref> in mind, we make the following decompositions. 


For i=1, …, k, we define E'_, i∈ C^∞_() and _i ∈ C^∞_() by requesting that E'_,i is supported on D^χ_[_i](3δ) and that 

    ∑_i=1^k E'_, i = E'_,   ∑_i=1^k _, i = Φ'_,
      _ _, i = E'_osc, i.


The functions Φ'_i are well-defined by Lemma <ref>.  By combining Definition <ref> with Lemma <ref>, we get global estimates for _.  
  

The following estimates hold. 
 

  * 
	
  * _, i : C^j_( , , e^-m| || - _i|)≤ C(j).
	
  * _, i : C^j_(D^χ_[_i](3δ),  )≤ C(j)/m.
	 

  * _osc : C^j_( ,  )≤ C(j).


  * _ : C^j_(D^χ_[_i](3δ),  )≤ C(j)/m. 
 


(i) follows directly from applying Lemma <ref> to E'_, i, using Lemma <ref> and Schauder regularity for the higher derivative estimates.  
For small k, (ii) follows from (i).  On the other hand, for k large enough in absolute terms, Lemma <ref>.(iv) implies that for all i, j ∈{1, …, k}, |_j - _i| > 3|j-i|/4k.  Using this with part (i) above, we estimate

    _osc : C^j_( ,  )   ≤  C(j) sup_∈∑_i=1^k e^-m| || - _i|
       ≤ C(j) ∑_l=0^k-1 e^- 3m/4k l 
       ≤ C(j),

where we have used Assumption <ref>.  This completes the proof of (ii).

Now fix some i∈{1, …, k}.
As in part (i), we may assume that for all i, j ∈{1, …, k}, |_j - _i| > 3|j-i|/4k.  Using the definitions and (i) above, 

    _ : C^r_(D^χ_[_i](3δ),  )   ≤_, i: C^r_(D^χ_[_i](3δ),  )
       +∑_ j≠ i_, j : C^r_(D^χ_[_i](3δ),  )
       ≤C(r)/m + C(r)∑_ j≠ i_, j : C^r_(D^χ_[_i](3δ),  )
       ≤C(r)/m + C(r) ∑_j≠ i e^-m| _j - _i|
       ≤C(r)/m+ C(r) ∑_l=1^k e^-3m/4kl 
       ≤C(r)/m,

where we have used Assumption <ref>.

We culminate our understanding of  with the following estimates.  
  

The following hold. 


  * : C^j_( ,  ) ≤ C(j). 

  * For i∈{ 1, …, k}, : C^j_( D^χ_[_i](3δ),  )≤ C(j)/m.



Because of the estimates on _ established in Proposition <ref>.(ii), it is enough to prove the estimate (i) for  Φ'_.  By Proposition <ref>, _ is supported on D^χ_L_par(3/m).  Fix i∈{1, …, k}.  We first establish the estimate on Ω[_i;m].  
Equation (<ref>) shows that there,

    _ =  [τ'_i/2; _i]  -_.
 
Note that the left hand side is smooth and the discontinuities on the right hand side cancel.  Using that __ = E'_ from (<ref>), on  Ω[_i;m] we have (where =   [ _i] is as in <ref>)

    ∂^2_ _ + 2/m^2^2( /m+ _i)  _ =  E'_.

On a neighborhood of ∂Ω[_i;m], we have that _ = 0 from Definition <ref>.  This combined with estimates on  from Lemma <ref> implies that |_|<C and | ∂_  Φ'_|<C on ∂Ω[_i;m].  Using this as initial data for the ODE and bounds of the inhomogeneous term from Lemma (<ref>) yields the C^2 bounds in (i).  Higher derivative estimates follow inductively from differentiating (<ref>) and again using Lemma <ref>.  This establishes (i) on Ω[_i;m].  The proof of the estimate (i)  on Ω[_i;m]∖Ω'[_i;m] follows in a similar way using (<ref>) but is even easier since there _ = 0, so we omit the details.

As in the proof of (i), Proposition <ref>.(iii) implies it is sufficient to prove the estimate in (ii) for _.
By Lemma <ref>.(iii), _ satisfies

    ∂^2_  _ +  2/m^2^2( /m+ _i)_+
    2/m^2  [ ^2 ] ( /m+_i) _ =    E'_.

The C^2 bounds in (ii) follow in a similar way as above, by using Lemma <ref>.(iii)-(iv) to estimate the initial data on ∂ D^χ_[_i](3δ), estimates on E'_ from Lemma <ref>.(iv), and estimates on [ ^2 ] and _ from Lemma <ref>.(iii) and (i) above.  Higher derivative bounds follow inductively from  differentiating (<ref>) and using Lemma <ref>.(iii) and Lemma <ref>.(iv).
 









§ MATCHED LINEARIZED DOUBLING (MLD) SOLUTIONS









 §.§ Mismatch and the spaces [L] and [L]

ab


 

We define a vector space [L] and 
given φ as in Definition <ref> with τ>0, the mismatch _L φ of φ, by

    _L φ : = ( _p φ)_p∈ L ∈ [L]:= ⊕_p∈ L[p], 
    
     where   _p φ : =  ( _p(p) + τ_p log (τ_p/2), d_p _p) ∈[p]:=⊕ T^*_p .




Among all LD solutions modulo [L], we are mainly interested in the ones which are well matched: 
 

We define a matched linearized doubling (MLD) solution modulo [L] of configuration (L, τ, w) 
to be some φ as in <ref> which moreover satisfies the conditions 
_Lφ = and τ_p >0 ∀ p ∈ L.


 

Given L = L[; m] as in <ref>, 
we define  

    _[L]  = span( (W_i)_i=1^k, ( W'_i )_i=1^k) 
       ⊂ C^∞_(),
    _[L] = 
    span( (V_i)_i=1^k, ( V'_i )_i=1^k) 
       =  
    {v∈ C^∞_(): v∈_[L] },

where for i∈{1, …, k} 
and with δ as in (<ref>), 
V_i , V'_i, W_i, W'_i ∈  C^∞_() are defined by requesting that they are supported on D^χ_L_i(2δ)  
and 
on D^χ_p_i(2δ) they satisfy   

    V_i    =V_i[_i, m]: = [ δ, 2δ; ^χ_p_i](  [ 1, 0 ; _i], 0),      
    W_i = W_i[_i,m]: =   V_i, 
    
     
    V'_i    =V'_i[_i, m]: =  [ δ, 2δ; ^χ_p_i]( [ 0, 1 ; _i ], 0),     
    W'_i = W'_i[_i,m]: =  V'_i.



Note that the last equality follows from the symmetries imposed. 
Note also that _[L] and _[L] are both 2k-dimensional 
with corresponding bases ( W_i, W'_i)_i=1^k and ( V_i, V'_i)_i=1^k.



We define a linear map _L : _[L] →_[L],
where _[L] was defined in (<ref>) and
_[L] is the subspace of [L] (recall <ref>)
consisting of those elements which are invariant under the obvious action of  on [L], by requesting that 
    _L (v) = (v(p),d_pv)_p∈ L∈_[L].
 


In Lemma <ref> below, we convert estimates established in Section <ref> for Φ on the cylinder into estimates for Φ on the sphere.  
Before doing this, we need the following lemma which compares the geometry induced by the metrics χ and g. 

 

 Let i ∈{1, …, k}.  There is a constant C>0—independent of m and k—such that:  
 

  * For any a, b ∈ (_i - 3/m, _i +3/m),

    ^2 a ∼_1+C/m^2 b.


  * When restricted to Ω[_i;m], 

    ^g_p_i∼_1+C/m_i  ^χ_p_i.
 

  * For large enough m and any ϵ≤δ/2,

    D^χ_L_i(ϵ/2)⊂ D^g_L_i( ϵ_i) ⊂ 
    D^χ_L_i(2ϵ).


  * If f∈ C^j(Ω), where Ω⊂ is a domain such that sup_p∈Ω|(p)|  ≤_k+1
	then 
	
    f: C^j( Ω, g)∼_C k^j/2 f: C^j(Ω, χ).
 
	


Fix i∈{1,…, k} and suppose _i - 3/m < a < b< _i + 3/m.  Trivially, ^2 b< ^2 a.  On the other hand, ∂( ^2 ) = -2 ^2 tanh, so by Grönwall's inequality,

    ^2 a    ≤(^2 b)   e^∫_a^b 2 tanh   d≤^2 b( 1+ C/m).

This completes the proof of (i).  (ii) follows easily from (i) and that g = ( ^2 )χ (recall (<ref>)).
(iii) follows from (ii) by taking m large enough.



(iv) follows by using that g = ( ^2 )χ in combination with part (iii) above and the fact (the second part of Proposition <ref>.(i)) that _k ∼_C k^-1/2.




For each i=1, …, k, V_i, V_i'∈ C^∞_() satisfy the following.


  * V_i: C^j_(,   )≤ C(j) and V_i' : C^j_(,  ) ≤ C(j).


  * _L is an isomorphism and ^-1_L≤ C m^2+β k^2+β/2(recall <ref>), where ^-1_L is the operator norm of ^-1_L: _[L] →_[L] with respect to the C^2, β(, g) norm on the target and the maximum norm subject to the standard metric g of . 




(i) follows easily from the bounds on  in Lemma <ref> and the uniform bounds on the cutoff in the  metric.  By the definitions above and (<ref>), it is easy to see that _L is invertible and that

    _L^-1( ( a_i , b_icos  d)_i=1^k) =  ∑_i=1^k a_i V_i + ∑_i=1^k b_i V'_i.

Combining (i) above with Lemma <ref>.(iv) to switch to the g metric (recall also (<ref>) and (<ref>)), we have

    V_i: C^j_(, g  )≤ C(j)k^j/2m^j,    V_i' : C^j_(, g  )≤ C(j)k^j/2m^j.

Recall that  is computed with respect to the C^2,β(,g) norm on _[L] and  
 the maximum norm on _[L], subject to the standard metric g of . Then (ii) follows by combining (<ref>) and (<ref>).




 §.§ The family of MLD solutions

ab


In this subsection we convert the LD solutions we constructed and studied in section <ref>  
to MLD solutions. 
We first have to choose the scale of the LD solutions so that we have approximate matching. 
By a heuristic argument which we omit we find that the overall scale  should be given by 

    := 1/m e^ζ_1 e^-ϕ(_1) = 1/m e^ζ_1 e^- m/F^ϕ_1,

where ζ_1 is an unbalancing parameter used to absorb error terms later. 
The continuous parameters of the construction are then 
:= (ζ_1, ) = (ζ_1, , ) ∈×^k-1×^k 
where we require

    |ζ_1| ≤_1,   ||_ℓ^∞≤_1/m,    | |_ℓ^∞≤_1/m,

where _1>0 will be fixed later and which we assume may be taken as large as necessary depending on k but independently of m.  

With the overall scale  having been chosen, 
we define the MLD solution 

    φ=Φ+
 
for some ∈[L] uniquely determined by the matching condition _Lφ=: 
By the definitions _Lφ=_L(Φ)+_L(). 
Using the invertibility of _L as in <ref>.(ii),  
the matching condition is equivalent then to 

    = -^-1_L _L (Φ).
 
To record in detail the dependence on the continuous parameters we have the following. 



We assume  is given as in (<ref>).
Let ϕ=  ϕ[: k,m], Φ = Φ[: k,m],  
and τ_i' = τ_i' [ : k] 
be as in <ref>.  
We define then 
= [;  m] 
by <ref>, 
an MLD solution 
φ = φ[ ; m] 
of configuration 
(   L   ,   τ  ,    w   ) (recall <ref>) 
by <ref> and <ref>, 
where L=L[; m],  
= [ : k ] (recall <ref>), 
τ= τ[ ; m] : L[; m] →_+ 
is -invariant satisfying τ_i = τ_1 τ'_i for i∈{1, …, k },   
and w= w[ ; m] :=. 
Finally we define 
=[; m] = (μ_i[; m])_i=1^k∈^k 
and 
'='[; m] = (μ'_i[; m])_i=1^k∈^k 
by 
= ∑_i=1^k τ_i μ_i V_i + ∑_i=1^k τ_i μ'_i V'_i 
which also implies 
w = ∑_i=1^k τ_i μ_i W_i + ∑_i=1^k τ_i μ'_i W'_i.  





Let φ be as in <ref>.  The equation _L φ = is equivalent to the equations

    0    = m/ F^ϕ_1( e^-∑_l=1^i-1σ_l - 1)+  (p_i)/τ'_i+ μ_i +ζ_1 + log(9/2τ'_i)- log_i 
    
     
    0    = 1/τ'_i∂/∂(p_i)  +  m/2ξ_i + μ'_i + 1/2tanh_i

for i=1, …, k.



From Definition <ref>, the condition that _Lφ = is equivalent to the conditions that

    1/τ_p_i_p_i(p_i) + log( τ_p_i/2) = 0,     1/τ_p_i∂_p_i/∂(p_i)= 0,      (i=1, …, k).

By Definitions <ref>, <ref> and <ref>, (<ref>), and Lemma <ref>, we have on D^χ_p_i(3δ)

    1/τ_i_p_i   = 1/τ'_iΦ - G_p_i+ μ_i V_i + μ'_i V'_i + ( G_p_i - G^^2∘^g_p_i).

Expanding Φ =  + + and using expressions for  and  from Definitions <ref> and <ref> gives 

    1/τ_i_p_i   = 
    1/τ'_i_i - [ logδ, 0; _i]
    +1/τ'_i+μ_iV_i + μ'_i V'_i +( G_p_i - G^^2∘^g_p_i).

Evaluating at p_i, using that V(p_i) = 1, adding log(τ_p_i/2) to both sides, using (<ref>)  to see that ϕ(_i)/τ'_i = m/F^ϕ_i, and using Lemma <ref> shows that the vertical matching equation is equivalent to

    0 = m/F^ϕ_i+ Φ'(p_i) /τ'_i + μ_i + log( τ'_i τ_1/2δ) - log_i.

Simplifying m/F^ϕ_i using (<ref>) and expanding the last term using (<ref>) gives (<ref>).

Next, note using the definition of _i in <ref>, the vertical balancing equation and (<ref>)

    1/τ'_i∂_i/∂(_i)    = ϕ(_i)/τ'_iF^ϕ_i+ - F^ϕ_i-/2= m/2F^ϕ_i+ - F^ϕ_i-/F^ϕ_i = m/2ξ_i.

(<ref>) follows from this decomposition, (<ref>), and Lemma <ref>.


[cf. <cit.>]

We fix some α>0 which we will assume as small in absolute terms
as needed.  We also fix some β∈(0,1) and γ∈ (1,2) 
satisfying 1-γ/2>2α and (1-α) (γ-1)>2α, for example γ=3/2.
We will suppress the dependence of various constants on β and γ.





For each p∈ L we define
δ_p'=τ_p^ 
where 
α is as in Convention <ref>.  For each p∈ L, there exists a unique i∈{1, …, k} such that p∈ L_i.  
Define then δ_p = (_i ) δ.
Define also 

    τ_min   =min_p∈ Lτ_p,   τ_max=max_p∈ Lτ_p,    
    δ_min   = min_p∈ Lδ_p,   δ_min'=min_p∈ Lδ_p'=τ_min^.








For Φ as in <ref>, the following estimate holds.

    Φ : C^3, β_( ∖  D^g_L(δ'_min), g)≤ C (   (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2).



Recall from Definition <ref> that Φ =  + +. 
By Lemmas <ref> and  <ref>, we have 

    : C^3, β_( ∖ D^χ_L(δ'_min), χ)≤ C (δ'_min)^-3-β| logδ'_min | .

Recall from <ref> that _ (  + ) = - _ and 
Since _ = 0 on D^χ_L(2δ),
it follows from <ref>.(i) that

    _ (  + ):  C^3, β_(  ),   ) ≤ C.

From <ref>.(i), <ref> and <ref>.(ii), we have also

    +  :  C^0_( ,    ) < Cmk,

so it follows from the Schauder estimates that

    +  : C^3, β_(  , χ) ≤ C m^4+β k .

Now let Ω: = ∩{ || ≤_k + 3/m}.  After combining (<ref>) and (<ref>) and using Lemma <ref>.(iv), we conclude  

    Φ : C^3, β_( Ω∖  D^g_L(δ'_min), g)≤ C (   (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2).

It remains to verify the estimate on ∖Ω. 
On ∖Ω, note that =0 (recall <ref>) and also that = c for c bounded independent of k, which follows from <ref> and <ref>.(iii). Using this and the exponential decay of  away from  from <ref>, we conclude Φ : C^3, β ( ∖Ω, g) ≤ m^4+βk.



The next lemma will be important in controlling certain error terms in the fixed point theorem.


For Φ as in Definition <ref> and 1≤ j < i ≤ k, we have

    τ'_i/τ'_j = ϕ(_i)/ϕ(_j)( e^∑_l=j^i-1σ_l)  ∼_1+C/k 1.



The first equality follows from <ref>.(i).  
We have then

    logτ'_i/τ'_j   = logϕ(_i)/ϕ(_j) + ∑_l=j^i-1σ_l=  O(1/k) + O(k _1/m),

where the estimates follow from Proposition <ref>.(iii), Definition <ref> and (<ref>).



Given = (a_i)_i=1^k ∈^k, k≥ 2, we define ∈^k-1 by requesting that ()_i = a_i+1 - a_i, i=1, …, k-1.
It is useful to think of  as a discrete derivative of .



 
Let  be as in (<ref>) and φ = φ[; m] be as in <ref>.
For m large enough as in <ref> (depending on _1), the following hold:


  * = [; m] and (, ') = ([; m], '[; m]) depend continuously on .  


  * [; m] ∼_C(_1)[; m] and C(_1)>1 depends only on _1. 
 

  * (Matching estimates) There is an absolute constant C independent of _1 such that 
	
	
  * |ζ_1+μ_1 | ≤ C.
	
  * |   - F^ϕ_1/m|_ℓ^∞≤ C/m.
	
  * |+  2/m' |_ℓ^∞≤ C/m.
	

  * φ : C^3, β_(∖ D^g_L(δ'_min), g) ≤τ_min^8/9.

  * On ∖ D^g_L(δ'_min) we have τ_max^1+α/5≤φ.

  * For all p∈ L, (δ_p)^-2_p: C^2, β( ∂ D^g_p( δ_p),( δ_p)^-2 g)≤τ_p^1-α/9.
	



 The continuity of the parameter dependence of  and  on  then follows from Definition <ref>, 
Proposition <ref> and Proposition <ref>.  

We next prove (ii).  For convenience in this proof, denote ϕ =(Φ[: k, m])_ and = ( Φ[: k,m])_ and = [:k], ' = [': k] (recall <ref>).  From Definition <ref>, 

    τ_1[ ;m]/τ_1[; m]   = e^ζ_1 e^ϕ'('_1) - ϕ(_1).

From  <ref>.(ii), Proposition <ref>, Proposition <ref>.(i), and <ref>, we have 

    |ϕ'('_1) - ϕ(_1)|   =m| F^ϕ_1- F^_1|/F^_1 F^ϕ_1≤ C _1.

This establishes (ii).
We next prove (iii).  
Taking i=1 in (<ref>) we obtain 

    μ_1 + ζ_1=-  log( 9/2) - (p_1) + log_1.

In conjunction with Proposition  <ref>.(ii), this proves (iii).(a).  
Now suppose i≥ 2.  
Subtracting the instance of (<ref>) evaluated at i from the instance evaluated at i-1 gives

    0    = μ_i-1 - μ_i+  m/ F^ϕ_1(  σ_i-1 + O ( ^2/m^2) )- log(τ'_i/τ'_i-1) 
    - (p_i)/τ'_i + (p_i-1)/τ'_i-1 + log_i/_i-1.

Multiplying through by F^ϕ_1/m and rearranging, we find

    F^ϕ_1/m(  μ_i-1-μ_i)+  σ_i-1   =  F^ϕ_1/m( logτ'_i/τ'_i-1 +(p_i)/τ'_i - (p_i-1)/τ'_i-1 - log_i/_i-1) 
    + O( _1^2/m^3).

We estimate the right hand side of (<ref>):
by Lemma <ref>, Proposition <ref>, and Proposition <ref>, 

    F^ϕ_1/m|logτ'_i/τ'_i-1| ≤C/mk^2,
       F^ϕ_1/m| log_i/_i-1| ≤C/mk,
       F^ϕ_1/m| (p_i)/τ'_i  - (p_i-1)/τ'_i-1|    ≤C/mk.

This completes the proof of (iii).(b).

Multiplying (<ref>) by 2/m and rearranging, we estimate that for 
i∈{1, …, k} we have  

    |  ξ_i+2/mμ'_i| ≤2/τ'_im|∂/∂(p_i)| +tanh_i/m.

(iii).(c) then follows from <ref> with a constant C depending on k.  

Next, since (V_i)∩(V_j) = ∅ when i≠ j and likewise for the functions V'_i, we estimate

    ∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i : C^3, β_( ,    )   ≤max_i=1, …, k( τ_iμ_i V_i : C^3, β_( ,    ) +τ_iμ'_iV'_i : C^3,β_( ,   )).

It follows from (iii) above and (<ref>) that
 
    |τ_i μ_i| ≤ C _1 ,    i=1, …, k.
 
 Therefore, using Lemma <ref> to estimate the norm of V_i, we find
 
    max_i=1, …, kτ_iμ_i V_i : C^3, β_( ,    )≤ C _1  .

 By (iii) above, we have |τ_i μ'_i | ≤ C _1 k^-2.  Using Lemma <ref> to estimate the norm of V'_i, we get

    max_i=1, …, kτ_iμ'_iV'_i : C^3, β_( ,  )≤ C_1 .

By combining (<ref>)and (<ref>) and switching to the g metric, we estimate using Lemma <ref>.(iv)

    ∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i : C^3, β_( , g )≤ C _1 k^3+β/2 m^3+ β.

Recalling from (<ref>) that φ = Φ+  ∑_i=1^kτ_iμ_i V_i + ∑_i=1^k τ_iμ'_i V'_i and combining (<ref>) with Lemma <ref> yields

    φ : C^3, β( ∖ D^g_L(δ'_min), g)≤ C(   (δ_min')^-3-β | logδ_min' | k^3+β/2 + m^4+βk^5+β/2+ _1 k^3+β/2m^3+β)τ_1.


From Definition <ref> and Lemma <ref>, δ'_min = τ_min^α∼_C τ^α_1.  (iv) then follows from the above by taking m large enough. 
For (v),
from (<ref>), (<ref>), Lemma <ref>.(ii) and Definition <ref>,

    : C^0(  ∖ D^g_L(δ'_min))   ≤α Cm k  
    ∑_i=1^k τ_iμ_i V_i + ∑_i=1^k τ_iμ'_i V'_i : C^0_sym(, g)   ≤ C_1 τ_1
     : C^0( , g)   ≤ C.

It is easy to see from  <ref>.(ii) and definition <ref> and that there is an absolute constant c>0 such that >  cmk, so (v) follows from <ref> and <ref> by taking α small enough and m large enough.  

Finally, let i∈{1, …, k}.  By <ref>, <ref> and <ref>, on D^g_p_i(δ_p_i), _p_i satisfies

    _p_i   = τ_1(  + ) + τ_i' τ_1 ( G_p_i - G^^2∘^g_p_i) + ∑_i=1^k τ_iμ_i V_i + ∑_i=1^kτ_i μ_i' V'_i
       := (I) + (II)+(III).

By <ref> and <ref>, (I): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ Cτ_1 mk^2+β.  By Lemma <ref>, G_p_i - G^^2∘^g_p_i satisfies the equation '( G_p_i - G^^2∘^g_p_i) = 0 and the C^0 estimate 

    G_p_i - G^^2∘^g_p_i : C^0( D^g_p_i(δ_p_i), g) < C|log_i|
    ≤ C k,

where the last estimate follows from <ref>.(i).  
Therefore, by the Schauder estimates, we have 


(I): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ C τ_1 k.  
Additionally, by (<ref>), it follows that 
(III): C^2, β( ∂ D^g_p_i(δ_p_i), (δ_p_i)^-2 g)≤ C _1 k^2+β/2τ_1.  
(vi) then follows by taking m large enough.





















§ MAIN RESULTS WITH NO NECKS AT THE POLES OR THE EQUATORIAL CIRCLE




 §.§ Initial surfaces from MLD solutions

ab


In this subsection we discuss the conversion of 
the MLD solutions (constructed in the previous subsection) 
to  initial surfaces.  
We also discuss the mean curvature and the linearized equation on 
the initial surfaces constructed. 
These steps were carried out under generous assumptions for the 
MLD solutions in <cit.> and therefore we only 
quote the results here and confirm that our MLD solutions 
satisfy the required conditions (see Lemma <ref>).  





For each p∈ L,
the disks D^g_p(9δ_p) are disjoint for different points p∈ L.


Follows from <ref>, Proposition <ref> and Assumption <ref> by taking m large enough. 




Let φ[;m] be as in <ref>.  The following hold. 


  * τ_max is small enough in absolute terms as needed.


  * ∀ p∈ L we have 9 δ_p' < τ_p^α/9< δ_p. 


  * τ_max≤τ_min^1-/9.  


  * ∀ p∈ L we have 
(δ_p)^-2 _p 
: C^2,β( ∂ D^g_p(δ_p)    ,  (δ_p)^-2 g ) ≤τ_p^1-/9.


  * φ:C^3,β_ (  ∖_q∈ LD^g_q(δ_q')      ,   
g   )  ≤τ_min^8/9.


  * On ∖_q∈ LD^g_q(δ_q') we have
τ_max^1+α/5≤φ.  


  * _L: _[L]→_[L] is a linear isomorphism and
	δ_min^-4 τ_max^α  ≤ 1.



(i)-(iii) follow from <ref> and <ref> by taking m large enough.  (iv)-(vi) follow from Lemma <ref>.(v)-(vii).  Finally, (vii) follows from <ref>.(ii).




Given a -symmetric MLD solution φ of configuration (L, τ, w)  as 
in <ref>, we modify it to 
∈ C^∞_(∖ L)
as in <cit.>. 
By using cut-off functions then we define as in 
<cit.> 

    φ_init
    =
    φ_init[L,τ,w] 
    :∖_p∈ L D_p(τ_p)→[0,∞),

and then as in 
<cit.> 
the initial smooth surface M[L,τ,w] as the union of the graphs of 
±. 
Recall however that our MLD solutions and their configurations 
(defined in <ref>) 
are parametrized by  (which determines also k) and m. 
Because of this we introduce the notation 

    M= M[[ ]] = 
    M[[ ; m ]] := M[ L, τ, w],    where  
    L=L[[]] = L[[ ; m ]] 
    := L[  [ : k ]  ; m],
 
τ= τ[ ; m], 
and w= w[ ; m],  
are as in <ref>.  
Note that the double brackets are introduced to distinguish from earlier notation. 
Note also that as usual the value of k is implied by  and we may not mention m 
when it is implied by the discussion. 

In the rest of this section 
we assume throughout that  satisfies (<ref>), (k,m) ∈^2 is as in Assumption <ref>, 
and m is large enough in terms of _1 as needed.
Now that the initial surfaces have been defined, 
we need to discuss their mean curvature, the linearized operator on them, and the nonlinear terms in a small perturbation. 
All these have been studied in <cit.> and so we only need to quote a definition and three basic results. 
Note that  was defined in (<ref>).  
Note also that Convention <ref> and Lemma <ref> imply 
all the requirements for the applicability of 
<cit.>  
so we do not mention them in <ref>. 





For k∈, ∈(0,1), 
∈,
and Ω a domain in  or
an initial surface M as above, 
we define
u_k,,;Ω
:=
u:C^k,(Ω ,ρ,g,ρ^),
where 
ρ:= ^g_L ∘ 
when Ω⊂
or Ω⊂ M, 
and g is the standard metric on , 
or the metric induced on M by the standard metric on ^3(1).





The modified mean curvature 
H-∘
on an initial surface M=M[[]] as above 
is supported on 
^-1(_p∈ L( D_p^g(3δ'_p) ∖ D_p^g(2δ'_p)  )).
Moreover it satisfies the estimate 

    H - ∘ _0,β,γ-2;M≤ τ_max^1 + α/3 
    .






If M=M[[]] is an initial surface as above,  
there exists a linear map 
_M: C^0,β_(M) → C^2,β_(M) ×_[L] such that if 

    _M E:= (u,w) ∈ C^2,β_(M) ×_[L],
 
then 

    u= E+ w ∘.
 
Moreover the following hold.


  * u_2,β,γ;M≤  C(b)   δ_min^-2-β    E _0,β,γ-2;M.


  * w  : C^0,β(,g)≤  C  δ_min^γ-2-β  E_0,β,γ-2;M.


  * _M 
depends continuously on the parameters of .



Given ∈ C^1_(M), we define the normal perturbation of ,  I_: M →^3(1),  by

    I_(x)=exp_x( (x) ν(x) ),

where ν:M→ T^3(1) is the unit normal to M. 
If  is sufficiently small, the normal perturbation M_: = I_(M)  is an embedded surface.  
Moreover, M_ is invariant under the action of  on the sphere ^3(1).

We recall the following estimate (<cit.>) on the nonlinear terms of such a perturbation.



If M is as in <ref> and 
∈ C^2,β_(M) 
satisfies _2,β,γ;M ≤ τ_max^1+α/4,  
then M_ is well defined as above and
is embedded.
If 
H_ is the mean curvature of M_ pulled back to M by I_ 
and H is the mean curvature of M, then we have 

    H_-  H -  _0,β,γ-2;M ≤  C  _2,β,γ;M^2.





 §.§ The main theorem with no bridges at the poles or the equatorial circle

ab



For the purposes of the fixed point theorem, we will need to fix a reference initial surface 
and pull back perturbations of the initial surfaces we consider to the reference surface via appropriate diffeomorphisms:  



There exists a family of diffeomorphisms _ : M[[]] → M[[]], 
where 
= (0, …, 0 )∈^2k,  
satisfying the following: 


  * _ depends continuously on .

  * _ is covariant under the action of .
 

  * For any u∈ C^2, β(M[[]]) and E∈ C^0, β(M[[]]) we have the following equivalence of norms: 

    u∘_ _2,β,γ;M[[]]  ∼_2    u _2,β,γ;M[[]]  ,
          E∘_ _0,β,γ-2;M[[]]  ∼_2    E _0,β,γ-2;M[[]]  .





As a preliminary step, we construct a family of diffeomorphisms '_L_par:→ which depend smoothly on L_par and covariant under the action of .
For ease of notation, denote the positive -coordinates of the circles L_par[[]] by  and likewise the coordinates of the circles in L_par[[]] by '.  We define '_L_par by requesting the following:


  * '_L_par is rotationally invariant in the sense that '_L_par((, θ)) depends only on . 

  * On D^χ_L_par[_i](5 δ), we have '_L_par( (, θ) ) = ('_i - _i + , θ).


  * On ∖ D^χ_L[[]](5δ), '_L_par we have

    '_L_par( (, θ)) = (f_L_par(), θ)
 
 for a suitably chosen function f_L_par.   

By choosing f_L_par carefully, we can ensure that '_L_par depends smoothly on L_par, hence on  and is close to the identity in all necessary norms.  Next, we use '_L_par to define _ by requesting the following.



  * ∀ p∈ L[[]] we define 
_ to map Λ_: = M[[]] ∩Π^-1_( D^g_p_i(δ'_p_i)) onto 
Λ_ : = M[[]] ∩Π^-1_( D^g_q_i(δ'_q_i)) 
where q_i = '_L_par(p_i).  
On Λ_, _ satisfies

    _∘ Y_∘Π_, p_i = Y_∘Π_, q_i∘_,

where Y_ (and similarly for Y_ is the conformal isometry from Π_, q_i(Λ_) equipped with the induced metric from the Euclidean metric 
. (τ[, m])^-2 g |_p_i, to  {∈ [-ℓ_, ℓ_]}⊂ (, χ), and

    _: 
    {∈ [-ℓ_,  ℓ_ ] }→{∈ [-ℓ_,  ℓ_ ]}
 
is of the form 

    _ (,θ) =
    (  ℓ_  /  ℓ_ ,  θ ),
 
where the ambiguity due to possibly modifying the θ coordinate by adding a constant is removed by the requirement that _ is covariant with respect to the action of .  

  * We define the restriction of _ on M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p)) to be a map onto M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p)) which preserves the sign of the  coordinate and satisfies 

    Π_∘_ = '_L_par∘Π_.


  * On the region M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (2δ'_p))∖ M[[]] ∩Π_^-1(∖_p∈ L[[]] D^g_p (δ'_p)) we apply the same definition as in (2) but with '_L_par appropriately modified by using cut-off functions and ^g_L[[]] so that the final definition provides an interpolation between (1) and (2).  

By construction, _ satisfies (i) and (ii).  
To check the uniform equivalence of the norms, 
first observe from the parametrization of the catenoid (<ref>)
and the uniform equivalence of the τs from Lemma <ref>.(ii), it follows that 

    ℓ_∼_1+ C(_1)/mℓ_.

Moreover, from Lemma <ref>.(ii), 
[; m] ∼_C(_1)[; m]. 
From this, it follows from arguing as in Lemma 4.13 of <cit.> that when m is large enough in terms of _1, the estimates in (iii) hold.  


For the convenience of the reader and to motivate the fixed point map  (see <ref>) 
used in the proof of the main theorem <ref> 
we recall the main steps of the construction 
omitting smallness conditions and the precise ranges of the parameters: 

 

Step 1: Unit RLD solutions:
Based on Lemmas <ref> and <ref> we construct in Proposition <ref> a unit 
RLD solution [_1; ] for given _1 ∈ (0, _root) 
and 
= (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^). 


Step 2: Unit RLD solutions with prescribed number of jumps and flux ratios:
Based on Proposition <ref> we introduce Notation <ref> to describe the 
RLD solutions which are smooth at the poles. 
These solutions are denoted by 
[: k] 
and are determined uniquely by the number of jumps k and 
the corresponding 2k-1 flux ratios 
= (, ) ∈^k-1×^k. 
Their jump latitudes are denoted by = [: k ]∈^k_+.  

 
Step 3: Normalized -symmetric LD solutions: 
Based on Lemmas <ref> and <ref> we construct in Lemma <ref> 
an LD solution Φ [ : k,m] 
of configuration (  L= L[ [: k]   ; m] ,  τ'   ) 
for given k,m∈ and 
= (, ) ∈^k-1×^k. 
Φ = Φ [ : k,m] is normalized by the condition τ'_1=1 and 
is characterized by the form of the configuration (recall <ref>) and the requirement that 
ϕ = ϕ[: k,m]:=Φ_ is a multiple of [ : k]. 
Note that the 
τ_i' = τ_i' [ : k]'s 
do not depend on m. 

 
Step 4: -symmetric MLD solutions: 
In this step we first choose a suitable overall scale τ_1 in 
(<ref>) for our LD solutions,  
and then modify them by adding  as in <ref> to obtain MLD solutions 
φ[ ; m] (see <ref>), 
where 
:= (ζ_1, ) = (ζ_1, , ) ∈×^k-1×^k = ^2k 
involves an extra parameter ζ_1 used in unbalancing related to the overall scale τ_1. 

 
Step 5: -symmetric initial surfaces: 
In this step the MLD solutions are converted to initial surfaces by using the process 
developed in <cit.>. 

 
Step 6: -symmetric minimal surfaces: 
In this remaining step we use a fixed point argument 
to perturb to minimality one of the initial surfaces constructed earlier for given k,m.  





There is an absolute constant _1>0 such that given (k,m)∈^2 with m large enough in terms of _1 and k, 
there is ∈^2k satisfying (<ref>) such that φ[ , m] satisfies the conditions of Lemma <ref> 
and moreover there is ∈ C^∞(), where : = M[[   ; m ]], 
such that

    _2, β, γ, ≤_1^1+ α/4,

and further the normal graph _ is a genus 2mk -1 embedded minimal surface in ^3(1) 
which is invariant under the action of  and has area Area(_)→ 8 π as m→∞.  







 
The proof is based on finding a fixed point for a map 
: B → C^2, β_sym(M[[]])×^2k 
we will define shortly, 
where 
B ⊂ C^2,β_( M[[]] )×^2k 
is defined by

    B := 
    {  v∈ C^2,β_(M[[]]):v_2,β,γ;M[[]] ≤ τ_1[;m]^1+α }× [-_1,_1]×[-_1/m,_1/m]^2k-1.

To motivate the definition of , suppose (v, ) ∈ B.  
Use Proposition <ref> to define (u, w_H):= - _M[[]]( H - w∘Π_).  
Define also ∈ C^2, β_(M[[]]) by : = v∘^-1_ + u.  
We then have: 


  * u + H = (w+w_H)∘Π_.

  * By Lemma <ref> and Proposition <ref>, 
	
    w_H: C^0, β( , g) +_2, β, γ; M[[ζ]]≤^1+α/4.

Using <ref> again, define (u_Q, μ_Q): = - _M[[]](H_- H - ).  By definition,

  * u_Q+H_ = H +  +w_Q ∘Π_.By <ref>, we have the following estimate on the quadratic terms:

  * w_Q : C^0, β( , g) +  u_Q_2, β, γ; M[[]]≤^2+α/4.By combining the above, we see

  * ( u_Q - v∘^-1_) + H_ = (w+w_H+w_Q)∘Π_.  


We then define

    (v, ) = ( u_Q ∘_, ζ_1+_1,     - F^ϕ_1/m , +2/m'
    ),

where ∑_i=1^k τ_i_i W_i + ∑_i=1^kτ_i '_i W'_i = w+w_H+ w_Q. 

We are now ready for the fixed-point argument.  Clearly B is convex.  Let β' ∈ (0, β).  By the Ascoli-Arzela theorem, the inclusion B ↪ C^2, β'_sym( M[[ ]]) is compact.  Moreover, by <ref>.(iii) and item (4) above, when m is large enough, (B) ⊂ B.   

The Schauder fixed point theorem implies there is a fixed point (, ) of .  By inspection of the defining formula for , this implies that _i = '_i = 0 for i=1, …, k, hence + _H+ _Q = 0 and also = _Q ∘ F_.  By (5), we conclude the minimality of _.  

The smoothness follows from standard regularity theory and the embeddedness follows from Lemma <ref> and (4) above.  The genus follows because we are connecting two spheres with 2km bridges.  Finally, the limit of the area as m→∞ follows from the bound on the norm of  and from the estimates (cf. <cit.>) on the function φ_init[  L[; m], , ] used to construct the initial surfaces. 





As k→∞, the sizes of the catenoidal bridges on each minimal surface in <ref> tends to become uniform in the following sense: 
given k∈ and ∈^2k as in (<ref>), 
it follows from Lemma <ref> that 
lim sup_m→∞τ_max[]/ τ_min[] < C/k 
where C is a constant independent of  and k.  
Consequently

    lim_k→∞lim_m→∞_max/_min = 1,

where _max/_min := _max[]/_min[] 
is the ratio of the maximum size over the minimum size of the bridges associated with fixed point (, ) of  as in <ref>.



 

For simplicity in this article we have not attempted to determine explicit bounds 
for m in terms of k (recall <ref>) 
under which the conclusion of Theorem <ref> holds. 
It would be interesting to extend our results to include the cases that k≤  m 
with m large depending on  but independently of k. 
The case m small with large k would require new ideas as remarked in <cit.>. 














§ CONSTRUCTIONS WITH NECKS AT THE POLES OR THE EQUATOR


ab
In this section, we construct doublings of  whose configurations L contain the poles, 
or points equally spaced along the equator circle of , or both, 
in addition to points distributed symmetrically along 2k parallel circles in  as in the previous sections.  



 §.§ The case with necks at the poles


ab


Given = (_1, …, _k) as in <ref> and L[;m] as in <ref>, define

    = [; m] = L[;m]∪ L_pol⊂,

where L_pol = { p_N, p_S} =  ( p_N).
Given also a -symmetric function τ:[; m]→, we denote 
τ_i := τ(p_i) for i=1, …, k and := τ(p_N)  = τ(p_S).



Given = (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^) as in <ref>, 
k∈ and _1 ∈   (a_k+1, , a_k, ) (recall <ref>), 
we modify [_1; ] to [_1; ] by removing the last discontinuity in the sense that 
[_1; ] := [_1; ] on {∈ [0, _k+1]} and 
[_1; ] is smooth on {∈ [_k, ∞)}. 
By defining 
A_k = A_k[_1; ] and B_k = B_k[_1; ] by (<ref>) as in Lemma <ref> 
the last condition is equivalent to 

    := A_k  + B_k   on  {∈ [_k, ∞)}.



 



In Definition <ref>.(i), we required that RLD solutions ϕ be positive.  
This ensures that RLD solutions are meaningful for the linearized doubling apparatus when L consists of points equally spaced on 2k parallel 
circles—in particular that each RLD solution ϕ is the average of an LD solution 
Φ = φ[ L; τ] for some -invariant τ: L →_+ (recall Lemma <ref>).  
When L also contains the poles {p_N, p_S}, the presence of logarithmic singularities at {p_N, p_S} dictates that 
the expansion of the average of an -invariant LD solution 
Φ = φ[ L ; τ] contains a positive multiple of  (recall <ref> and <ref>.(ii)) on {∈ [_k, ∞)}.  
This is ensured by <ref>, (<ref>) and <ref>.




 
There exists _2>0 such that given k∈ and = (, ) ∈ℓ^1 ( ^) ⊕ℓ^∞( ^) satisfying ||_ℓ^1 + ||_ℓ^∞<_1/k (recall <ref>), A_k[_1; ] is a strictly decreasing function of _1 on [ a_k,  - _2/ k^2, a_k, ] which on this interval satisfies ∂ A_k [_1; ]/∂_1∼_C -k and A_k[ a_k, ; ] = 0, where C>0 is a constant independent of k.


Let k∈, = (, ) ∈ℓ^1( ^) ⊕ℓ^∞( ^) satisfy ||_ℓ^1 + ||_ℓ^∞<_1/k, and [_1; ] be as in <ref>, where _1 ∈ (a_k+1, , a_k, ).  By combining <ref> with <ref>  we find that

    F^_-(a_k, ) = F^[ a_k, ; ]_1-∼_exp(||_ℓ^1 + 3||_ℓ^∞) F^[ a_k, ; ]_ = 1/k + O ( 1/k^3).


 By a direct calculation, F^_-() =  + O(^2) for small .  We we conclude from the preceding in combination with the assumption that ||_ℓ^1+||_ℓ^∞<_1/k that 
 a_k,  - a_k+1, > C/k^2.
Assume now that _1 ∈ [ a_k, - _2/k^2, a_k, ] and that _2>0 is small enough that a_k, - _2/k^2> a_k+1,.
 From <ref> we have 

    A_k[_1; ] = (_k)( F^_+(_k) - F^_k+)(_k).

Differentiating (<ref>) with respect to _1, we find

    ∂ A_k[_1; ] /∂_1 = (I) + (II)   where
    
    (I) :=  ∂(_k)/∂_1( F^_+(_k) - F^_k+)(_k),   
    (II):=
    (_k) ∂/∂_1[ ( F^_+(_k) - F^_k+)(_k)].

Using (<ref>) and <ref> we find

    (II) = (_k)( - 2^2 _k (_k) ∂_k/∂_1
    - ∂ F^_-(_1)/∂_11+ξ_k/1-ξ_1( e^∑_l=1^k-1σ_l) - 
    F^_k+∂(_k) ∂_k/∂_1).

Using Corollary <ref>, <ref>.(i), and that F^_-() =  + O(^2) for small , we see that (II) ∼_C -k.  
To estimate (I), first observe from the estimate on (II) and that F^_+(_k) - F^_k+ = 0 when _1 = a_k, that

    ( F^_+(_k) - F^_k+) < Ck( a_k,  - _1) < C _2/k.


Recall now from <ref> that log( (_k)) = ∫_0^_k F^_+()d.  Differentiating under the integral sign, we find

    1/(_k)∂(_k)/∂_1 = - F^_k-∂_k/∂_1 -∑_i=1^k-1 F^_i ∂_i/∂_1  + 
     ∫_0^_k∂ F^_+()/∂_1 d.

Using Corollary <ref>, we estimate 

    ∂_i/∂_1 < C( 2 ^2 _i + ( F^_i-)^2)^-1  i ,    i=1, …, k,

hence by <ref> and <ref>.(i)

    F^_k-∂_k/∂_1 + ∑_i=1^k-1 F^_i ∂_i/∂_1 < Ck^2.

Using <ref>, when ∈ [_i, _i+1], i=1, …, k-1, we have

    ∂ F^_+/∂_1 = ( (_i)/())^21/21+ ξ_i/1 - ξ_1( e^∑_l=1^i-1σ_l) ∂ F^_-/∂_1(_1) +  ( (_i)/())^2(2 ^2 _i + (F^_i+)^2)∂_i/∂ F_1

Using <ref> and <ref>, we estimate |∂ F^_+/∂_1() |< k for ∈ [ 0, _k].  From this and the estimate that _k < Clog k (recall <ref>), we find

    ∫_0^_k∂ F^_+/∂_1() d < C k log k.

Combining with (<ref>), we have by estimating (<ref>) that 

    | ∂(_k)/∂_1| < C k^2.

Combining this with (<ref>) we find that  |(I)|< C k _2.  The result then follows from the estimates on (I) and (II) by taking _2 small enough.




  
Given k∈,  as in <ref>, and ∈ [0, _3/k), 
where _3>0 is a constant depending only on _2 and the constant C>0 in Lemma <ref>,
we define [ ,  : k] := [ _1; ], 
where _1 is uniquely characterized (recall Lemma <ref>) by the properties that 
_1 ∈ [a_k,  - _1/k^2, a_k, ] and A_k[_1; ] =.
Note that by <ref> [ , 0 :k] = [: k].
We also define B_k,  := B_k[,  : k] as in (<ref>).





Given [, : k] as in <ref>, we define _k∈ (_k, ∞) to be the unique root of F^_+ in (_k, ∞).  Note that _k is well defined by Lemma <ref>.





There is a constant _1>0 such that for all k∈ and all (, )= (, , )∈ℓ^1(^)⊕ℓ^∞( ^)×_+ with ||_ℓ^1+ ||_ℓ^∞ < _1 and ∈ [0,  _3/ k), 
=[, : k] satisfies the following.


  * F^_avg = 1/k +O(1/k^2) and ^2 _k ∼_C 1/k.  

  * 
	
  * F^_1-=2tanh_1 +O(1/k^3).
	
  * F^_i-+F^_i-1+ = 2(tanh_i - tanh_i-1) +O(1/k^2(k-i+1)) for i=2, …, k.
	
  * F^_k+=2(tanh_k- tanh_k) + O(1/k^2).
	

  * 1-(): C^0( {∈ [0, _k]}) ≤C/klog k and  |log(_i)/(_i-1)| ≤C/k for 2≤ i ≤ k.    If =, then also 1>(_1)> ⋯ >(_k).  


  * There is a constant C>0 depending only on _1 such that for any 1≤ i<j≤ k, 

    _j - _i    >  (1+C/k) j-i/2k  and  ^2 _i - ^2 _j ≤C(j-i)/k.




Since [_1; ] depends continuously on _1 on compact subsets of  and lim_↘ 0[ , : k] = [: k ], it follows from Proposition <ref> that for  sufficiently small (depending on k), the estimates in <ref> hold.  It remains to be seen that these estimates hold when ∈ [0, _3/k).  

To this end, note that on {∈ [_k, ∞)}, 

    F^_+()  = ∂ () + B_k, ∂()/() +  B_k, () 
    = ^2 /tanh (I) + (II) 
      where  
     
    (I) :=  B_k,  - / B_k, +  (  - ), 
         
    (II) := / B_k,  +  ( - ).

Evaluating (<ref>) at _k, taking ∈ [0, _3/k), and using that _k ∼_C log k, which follows from the fact that ^[: k]_k ∼_C log k (recall <ref>.(i)) by using the flux monotonicity and that _1<a_k,, we estimate that
 0<(I) < 2 and |(II)| < _3/k.  Combining this with (<ref>) and (<ref>), it follows that 

    ^2 _k ∼_Cexp(||_ℓ^1+ ||_ℓ^∞) F^_.

The rest of the proof proceeds in the same way as the proof of <ref>, so we point out only the differences.  The last equation in (<ref>) must be replaced with 

    F^_k+ = 2( tanh_k - tanh_k ) + ∫__k^_k( F^_-())^2 d.

It follows that tanh_k = 1 - O(1/k).  Estimating, as in the proof of <ref>, we find F^_ = 1/k + O(1/k).  The proofs of the remaining parts are nearly the same as the proof of <ref>, so we omit them.




 Let k∈ and = (, )∈^k-1×^k, '= (', ') ∈^k-1×^k.  Suppose that ||_ℓ^1+ ||_ℓ^∞< _1/k, |'|_ℓ^1+ |'|_ℓ^∞< _1/k, and ' ∈ [0,  _3/k), ∈ [0,  _3/k).  Let = [', : k] and ' = [ , ': k]. There is a constant C>0 independent of k such that: 

  * | ^' - ^|_ℓ^∞≤C/k(  | ' - |_ℓ^1+ |' - |_ℓ^∞+|'- |).  



  * max_1≤ i ≤ k| tanh'_i - tanh_i | ≤C/k(  | ' - |_ℓ^1+ |' - |_ℓ^∞+ |' - |).  




We first prove (i).  
As in the proof of <ref>, it suffices to prove

    | F^'_1 - F^_1 | ≤C/k(  |' -  |_ℓ^1 + |' - |_ℓ^∞).

Fix k∈, let _1 be as in <ref>, and consider the map

    ( F_1, , ) = (F^[_1; ]_+(_k) - F^_+(_k)) [_1; ](_k) (_k) + ,
 
where [_1; ] is as in <ref> and _1 is chosen so that F^_1 = F_1.  Recall from Remark <ref> that 

    = (_k)(_k) ( F^_+(_k) - F^_+(_k))
       + (_k)(_k)( - F^_+(_k) + F^_+(_k))

By combining (<ref>) with <ref>, we find that [_1; ] = [, : k] if and only if ( F_1, , ) = 0.
Now let (F_1, , ) ∈^-1({ 0}) be arbitrary.  Estimating the partial derivatives of  in a similar manner as in the proof of <ref> using <ref> to estimate the partial derivatives of _k, we estimate

    .∂/∂ F_1|_∼_C k,     | .∂/∂σ_i|_| ≤ C,     |. ∂/∂ξ_i|_| ≤C/k,     . ∂/∂|_ = 1.

By the implicit function theorem, locally around ∈^-1( { 0}), ^-1( { 0}) is a graph over ( , ) and moreover (abusing notation slightly),

    .∂ F_1/∂σ_i|_(, )   = - (. ∂/∂ F_1|_)^-1.∂/∂σ_i|_
    .∂ F_1/∂ξ_j|_(, )   = - ( .∂/∂ F_1|_)^-1. ∂/∂ξ_j|_
    .∂ F_1/∂|_(, )   = -( .∂/∂ F_1|_)^-1. ∂/∂|_.

(<ref>) follows from this and the estimates (<ref>).
(ii) follows from a similar, but omitted, calculation.





Given (, ) = (, , ) ∈^k-1×^k ×_+ as in <ref> and m as in <ref>, there is a unique -invariant LD solution (recall <ref>)

    Φ = Φ[, : k, m] := φ[ ; τ'],

characterized by the requirements that 


  * ϕ = ϕ[ ,  : k,m] : = Φ_ is a multiple of [, : k]

  * =  [  [ , : k]  ; m   ] (recall <ref>)

  * τ'_1 = 1 (normalizing condition).



Moreover, the following hold: 
 

  * For i∈{1, …, k} we have τ'_i =  ϕ(_i)/m F^ϕ_i.  Moreover τ'_i is independent of m and satisfies
	 τ'_i = [, : k](_i)/[, : k ](_1) e^∑_l=1^i-1σ_l.
	
	
  * ' = m/F^[, : k]_1 [, : k](_1).

  * ϕ = m/[, : k](_1) F^[, : k]_1[, : k].




The proof of the uniqueness part proceeds as in the proof of <ref>, so we only discuss the existence part.  As in the proof of Lemma <ref>, it follows that f: = ϕ - Φ_ is in C^∞_||(^)∩ C^1_||(), hence therefore f = c_odd + c_even for some c_odd, c_even≥ 0.  Since f∈ C^0_||(), c_odd =0. By the choice of τ' in (i)-(ii) above, it follows that f is smooth at the poles, hence c_even=0 and therefore ϕ= Φ_.  The proof of (iii) is exactly the same as in the proof of <ref>.




Let Φ[, :k, m] be as in <ref>.  Define ∈ C^∞_sym( ∖ L), ∈ C^∞_||( ), and , E' ∈ C^∞_sym() using Definitions <ref>, <ref> and <ref> with Φ[, :k,m] in place of Φ[: k,m]. 



While the defining formula for [, : k, m] is the analogous to the one for [: k, m], note that [, : k, m] is not extendible to C^∞_||() since [, : k] is not smooth at the poles. 




Given [;m] as in <ref>, define , ∈ C^∞_() by requesting that  is supported on D^g_L_pol(2δ)∖ D^g_L_pol(δ) and on 
D^g_p_N(2δ)

    =[m] = [ δ, 2δ; ^g_p_N] ( cos∘^g_p_N, 0 ),

and that = [m] :=.  
 We define 
 
    _ [ L_pol]    = span( ) ⊂ C^∞_ ( ),
    _ [ L_pol]    = { v∈ C^∞_() :  v ∈_ [ L_pol]}⊂  C^∞_ ( ), 
    _ [  ]    = _[ L] ⊕_ [ L_pol]⊂ C^∞_ ( ),
    _[ ]    = _[L] ⊕_[L_pol]⊂ C^∞_ ( ),

where _[ L], _[L] are as in (<ref>).  


By the symmetries, clearly _ [ L_pol] is spanned by .


For i=1, …, k, V_i, V'_i∈ C^∞_ () satisfy (i) of <ref>.  Moreover,

    : C^j_( ,   )≤ C(j).

Furthermore, _ is an isomorphism and ^-1_≤ C m^2+β k^2+β/2(recall <ref>), where ^-1_ is the operator norm of ^-1_: _[] →_[] with respect to the C^2, β(, g) norm on the target and the maximum norm subject to the standard metric g of . 


The estimates on V_i, V'_i follow exactly as in the proof of <ref>.(i).  The estimate (<ref>) follows the uniform bounds of the cutoff  in the  metric.  By the symmetries, _[L_pol] is one dimensional and _L_pol is an isomorphism. Moreover (abusing notation slightly) ^-1_L_pol( (1, 0)) = V_pol and it is easy to see that ^-1_L_pol≤ C m^2+β.

 It is clear that ^-1_ splits naturally, i.e. ^-1_ = ^-1_L + ^-1_L_pol, so the estimate on _ follows from the above and the estimate on _L established in <ref>.


We now convert the LD solutions constructed above to MLD solutions.  By heuristic arguments which we omit we find that the overall scale τ_1 should be given by

    τ_1:= 1/m e^ζ_1 e^-ϕ(_1) = 1/m e^ζ_1 e^- m/F^ϕ_1

and that the scale for the bridges at the poles should be given by 

    = [;m] : = e^F^_1/m B_k, 0,

where ζ_1 and  are unbalancing parameters used to absorb error terms later.  The continuous parameters of the construction are then : = (ζ_1, , ) = (ζ_1, , , ) ∈×^k-1×^k×, where we require

    (ζ_1, )   satisfy (<ref>)  and  | | ≤_1 log m/m,

where _1>0 will be fixed later and which we assume may be taken as large as necessary depending on k bud independently of m.
With the overall scale τ_1 having been chosen, we define the MLD solution

    φ = τ_1 Φ +

for some ∈_[ ] uniquely determined by the matching condition _Lφ=: 
By the definitions _Lφ=_L(Φ)+_L(). 
Using the invertibility of _L as in <ref>.(ii),  
the matching condition is equivalent then to 

    = -^-1_L _L (Φ).
 
To record in detail the dependence on the continuous parameters we have the following. 




We assume  is given as in <ref>.  Let ϕ = ϕ[, : k, m], Φ = Φ[ , : k,m] and τ' = τ'[: k] be as in <ref>.  We define then τ_1 = τ_1[ ; m] by (<ref>), an MLD solution φ = φ[; m ] of configuration ( , τ, w  ) (recall <ref>) by <ref> and <ref>, where = [;m], = [, : k] (recall <ref>), τ = τ[; m ] : [;m ] →_+ is -invariant satisfying τ_i = τ_1 τ'_i for i∈{1, …, k}, = τ_1 ', and w = w[ ; m]:=.  Finally we define = [; m] = (μ_i)_i=1^k ∈^k, ' = '[; m] = (μ'_i)_i=1^k ∈^k, and ∈ by = ∑_i=1^kτ_i μ_i V_i + ∑_i=1^kτ_i μ'_i V'_i + τ_pol, which also implies w = ∑_i=1^k τ_i μ_i W_i + ∑_i=1^k τ_i μ'_i W'_i +.





Let φ be as in <ref>.  
The equation _φ = is equivalent to the system of (<ref>) and (<ref>)   
for i=1, …, k, 
and the condition that (recall  (<ref>))

    0 = F^ϕ_1/m - 1 + F^ϕ_1/m( ζ_1 + log'/4m + 1) + F^ϕ_1/m B_k, /.



By Lemma <ref>, the equation _Lφ= is equivalent to the system of (<ref>) and (<ref>).    
Recall that τ_p_N =  = τ_1 m/F^ϕ_1 (_1).   
By <ref>, = L ∪ L_pol.  
The condition that _L_polφ = is therefore equivalent to the conditions that

    1/_p_N(p_N) +  logτ_1 '/2 = 0,     d_p_N_p_N = 0.

It is automatic from the symmetries that d_p_N_p_N = 0.  By Definitions <ref> and <ref> and <ref>.(iii), it follows that on D^g_p_N(3 δ),

    1/_p_N = F^ϕ_1 (_1)/ mΦ -  G^^2∘^g_p_N +  .

Expanding Φ = + + (recall <ref>), noting from <ref> that  vanishes on D^g_p_N(3 δ), using <ref> and <ref> to expand , and <ref>.(iii) to expand G^^2∘^g_p_N, we find that on D^g_p_N(3 δ),

    1/_p_N = B_k, / +  + F^ϕ_1 (_1)/m -  ( log 2 - 1)  -   +   .

Evaluating at p_N, and using that (p_N) = 1, and that (p_N) = 0, which follows from Lemma <ref>, we have 

    1/_p_N (p_N) = B_k, / - ( log 2 - 1)  +

Adding logτ_1 '/2 to both sides and using <ref> to expand τ_1, we find that (<ref>) is equivalent to

    0 =   - m/F^ϕ_1  +  ( ζ_1 + log'/4m + 1) +  B_k, /.

Multiplying through by F^ϕ_1/m completes the proof. 



 
Let  be as in (<ref>) and φ = φ[; m] be as in <ref>.
For m large enough (depending only on _1), the following hold:


  * = [; m] and = [; m] depend continuously on .  


  * [; m] ∼_C(_1)[; m] and C(_1)>1 depends only on _1. 
 

  * (Matching estimates) There is an absolute constant C independent of _1 such that 
	
	
  * |ζ_1+μ_1 | ≤ C.
	
  * |   - F^ϕ_1/m|_ℓ^∞≤ C/m.
	
  * |+  2/m' |_ℓ^∞≤ C/m.
	
  * |  -  F^ϕ_1/m| ≤ C log m/m.
	

  * φ : C^3, β_(∖ D^g_(δ'_min), g) ≤τ_min^8/9.

  * On ∖ D^g_(δ'_min) we have τ_max^1+α/5≤φ.

  * For all p∈ L, (δ_p)^-2_p: C^2, β( ∂ D^g_p( δ_p),( δ_p)^-2 g)≤τ_p^1-α/9.



The bulk of the proof is nearly the same as that of Lemma <ref>, so we only prove the essentially new estimate, (iii).(d).  Note that by <ref>, we have

    0 = F^ϕ_1/m - 1 + F^ϕ_1/m( ζ_1 + log'/4m + 1) + F^ϕ_1/m B_k, /.

 Using Lemma <ref>.(iii) to expand ' and , we find

    0 = F^ϕ_1/m - 1 + F^ϕ_1/m(-log m + ζ_1 + 1 +  + logF^_1 B_k, 0/F^ϕ_1 (_1)) + B_k,  F^ϕ_1/B_k, 0 F^_1 e^- ζ_1.

Expanding 

    B_k,  F^ϕ_1/B_k, 0 F^_1 e^- ζ_1 = 1 + B_k,  F^ϕ_1 - B_k, 0 F^_1/B_k, 0 F^_1 -  + O(^2),

we find the matching equation is equivalent to

    F^ϕ_1/m -  = F^ϕ_1/m(  log m + O(1)) -  B_k,  F^ϕ_1 - B_k, 0 F^_1/B_k, 0 F^_1 + O(^2).

The estimate now follows from (<ref>), <ref>, and <ref>.




There is an absolute constant _1>0 such that if (k, m)∈^2 with m large enough in terms of _1 and k, there is ∈^2k+1 satisfying (<ref>) such that φ[ ; m] satisfies the conditions of Lemma <ref> and moreover there is ∈ C^∞(), where : = M[[]] such that

    _2, β, γ, ≤^1+ α/4_1,

and further the normal graph _ is a genus 2km+1 embedded minimal surface in ^3(1) which is invariant under the action of  and has area Area(_)→ 8 π as m→∞.  



Given = (ζ_1, , ) ∈×^2k-1× as in <ref>, we denote = (0, …, 0) ∈^2k+1 and 

    M[[  ]] : = M[ [; m], τ[; m], w[; m]],   [[]] := [; m].
 

Define

    B    ={  v∈ C^2,β_(M[[]]):v_2,β,γ;M[[]] ≤ τ_1[; m]^1+α }× [-_1,_1]×[-_1/m,_1/m]^2k-1
       × [ - _1 log m/m, _1 log m/m]
     ⊂ 
    C^2,β_( M[[]] )×^2k+1.

The proof is the same in structure as the proof of <ref>; in particular, after carrying out steps (1)-(5) as in the proof of <ref>, we define

    (v, ) = ( u_Q ∘_, ζ_1+_1,   - F^ϕ_1/m ,  + 2/m',  -  F^ϕ_1/m_pol),

where ∑_i=1^k τ_i _iW_i + ∑_i=1^k τ_i '_i W'_i + _pol = w+w_H + w_Q.

It follows from Lemma <ref>.(iii) and the estimates on the norms of w_H and w_Q as in the proof of <ref> that ( B) ⊂ B, so we may apply the Schauder fixed-point theorem as in the proof of <ref>.




As k→∞, the sizes of the catenoidal bridges on each minimal surface in <ref> tends to become uniform in the same sense as in the case where there are no bridges at the poles (recall <ref>).  To see this, let k ∈ and ∈^2k+1 be as in <ref>.  By  <ref> and  <ref>, 

    [;m]/τ_1[; m] = '[;m] = m/F^_1 (_1) = F^_1 B_k, 0/F^_1 (_1) e^,

By <ref>, <ref>, and <ref>, it follows that

    lim_k→∞lim_m→∞_pol/ _1 =1,

where _pol/_1 is the ratio of the size of the bridges at the poles over the size of the bridges at the first jump latitude of a fixed point of  as in <ref>.  Then arguing   as in remark <ref> to compare with the sizes of the rest of the bridges away from the poles, we conclude 

    lim_k→∞lim_m→∞_max/_min = 1,

where _max/_min is the ratio of the maximum size over the minimum size of the bridges associated with a fixed point of  as in <ref>.













 §.§ The case with necks along the equator


ab


Here we construct doublings of  where  contains the equator circle.  Most of the construction is as before, so we outline the argument and describe in detail only those aspects which differ from the construction in Theorem <ref>.  To begin, we define an expanded class of RLD solutions.



Given k∈ and 0=_0 <_1<⋯<_k < ∞, we denote
= (_0, _1, …, _k)∈^k+1. 




We say ϕ∈ C^0_||( ) is a rotationally invariant linearized doubling solution if ϕ is as in <ref>, except ^ϕ in <ref>.(ii) may be as in either <ref> or <ref>; in the latter case we require that <ref>.(iii) holds for i=0, …, k.


From now on we assume ϕ is an RLD solution as in <ref> where ^ϕ is as in <ref>. 



Let ϕ be an RLD solution as in <ref>.  Define

    ^ϕ := ( F^ϕ_0-, F^ϕ_0+, F^ϕ_1-, …, F^ϕ_k+) ∈^2k+2_+,
    
    ^ϕ := ( F^ϕ_i)_i=0^k  ∈^k+1_+,
      ^ϕ := (σ^ϕ_i)_i=0^k-1∈^k, 
      ^ϕ = ( ξ^ϕ_i)_i=1^k ∈^k,

where for i=0, …, k,  j=0, …, k-1, and l = 1, …, k, 

    F^ϕ_i± := F^ϕ_± (_i),    
    F^ϕ_i := F^ϕ_+(_i)+F^ϕ_-(_i),    
    e^σ^ϕ_j = F^ϕ_j+1/F^ϕ_j,   ξ^ϕ_l = F^ϕ_l+ - F^ϕ_l-/F^ϕ_l+ + F^ϕ_l-.
 
We define ^ϕ: = (^ϕ, ^ϕ) ∈^k×^k and call the entries of ^ϕ the flux ratios of ϕ.   
If ^ϕ =  0 we call ϕ balanced.        
Finally we define

    F^ϕ_avg : = 1/2(k+1)|  ^ϕ|_ℓ^1 =  1/2(k+1)| ^ϕ|_ℓ^1.


Note that if ϕ is an RLD solution as in Definition <ref>, the assumption that ϕ∈ C^∞_||( ∖ L_par[]) implies that F^ϕ_0+ = F^ϕ_0-.  For this reason, we did not define ξ^ϕ_0. 

We omit the proof of the next result, which is a straightforward modification of the proof of <ref>.


Given F∈ (0, ∞) and 

    = (, )  = 
    (   (σ_i )_i=1^∞  ,   (ξ_j)_j=1^∞ )∈ℓ^1(^) ⊕ℓ^∞( ^)

satisfying | |_ℓ^∞ < 1/10, there is a unique unit RLD solution [F; ] satisfying the following. 


  * ^ is as in <ref> and F^[F; ]_+(_0) = F.

  * ^ = . |_k where k is the number of jump latitudes of ϕ and . |_k := (   (σ_i)_i=0^k-1, (ξ_i)_i=1^k ).

 Moreover, the following hold. 


 
 
  * k[F; ] is a nonincreasing function of F.  Further, for each  as above there exists a decreasing sequence {b_0, : = ∞, b_1, , b_2, , …} such that k[F; ] = k if and only if F ∈ [b_k, , b_k-1, ).  Moreover each b_k, depends only on . |_k (defined as above).


  * ^_1, …, ^_k are increasing smooth functions of F for fixed .


  * [F; ] is smooth at the poles if and only if F = b_k, for some k≥ 1. 


  * The restriction of [F; ] on any compact subset [0, ∞) depends continuously on F and .


We emphasize that the only substantial difference between RLD solutions = [F; ] as in <ref> and RLD solutions = [_1; ] as in <ref> is that while [_1;] coincides with  on {∈ [ -^_1, ^_1]}, the expansion of  on {∈ [ -^_1, ^_1]} with respect to the basis {, } contains a positive multiple of . 


 
 
Given k,  and .|_k as in <ref>, we define

    [: k] : = [ .|_k : k]: =  [ b_k, ; ],      = [: k] := [ .|_k :k] : = ^[: k].

By modifying Definition <ref>, Lemma <ref>, and Definition <ref> we also define for all appropriately small >0 RLD solutions
[ , : k] = [ _1; ]
which satisfy A_k[ _1; ] =.


Straightforward modifications of the respective statements and proofs of <ref> and <ref> provide versions of those results which hold for smooth at the poles RLD solutions  and  as in <ref>.  Furthermore, as in Lemma <ref> we convert each RLD solution  (or ) as in <ref> into an LD solution whose non-oscillatory part is a multiple of  (or ).  The only important difference is that (recall <ref>.(c)) these LD solutions are normalized so that τ'(p_0):= τ'_0 = 1.

These LD solutions are decomposed and estimated via obvious modifications of the machinery developed in Section <ref> used to estimate LD solutions Φ whose configurations L do not contain points on the equator circle of .




For constructions where L contains points on the equatorial circle but not at the poles of , the parameters of the construction are _eq =( ζ_0, ) = (ζ_0, , ) ∈×^k×^k, where we require

    |ζ_0| ≤_1,    ||_ℓ^∞≤_1/m,    ||_ℓ^∞≤_1/m.

In cases where L contains both points on the equator and at the poles, the parameters are
_eq =( ζ_0, , ) = (ζ_0, , , ) ∈×^k ×^k × where we require that

    (ζ_0, )   satisfies  (<ref>)  and    || < _1 log m /m.


Using the preceding, we modify the steps in Sections <ref> and <ref> construct MLD solutions φ[ _eq; m] and φ[ _eq;m] and in turn smooth initial surfaces M[[ _eq]] and M[[_eq]].  For the sake of brevity, we omit the proofs of the following and leave the routine modifications to the reader. 




There is an absolute constant _1>0 such that if (k,m)∈^2 satisfies Assumption <ref>, there is _eq∈^2k+1 satisfying (<ref>) 
 and moreover there is ∈ C^∞(), where : = M[[ _eq]] such that

    _2, β, γ, ≤_0^1+ α/4,

and further the normal graph _ is a genus (2k+1)m-1 embedded minimal surface in ^3(1) which is invariant under the action of  and has area Area(_)→ 8 π as m→∞.  




There is an absolute constant _1>0 such that if (k,m)∈^2 satisfies Assumption <ref>, there is _eq∈^2k+2 satisfying (<ref>)
 and moreover there is ∈ C^∞(), where : = M[ [ _eq ]] such that

    _2, β, γ, ≤_0^1+ α/4,

and further the normal graph _ is a genus (2k+3)m-1 embedded minimal surface in ^3(1) which is invariant under the action of  and has area Area(_)→ 8 π as m→∞.  












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