Unique asymptotics of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow
Sigurd Angenent Panagiota Daskalopoulos Natasa Sesum

Asymptotics (December 30, 2023)]
Unique asymptotics of ancient compact non-collapsed solutions
to the

3-dimensional Ricci flow 
Angenent]Sigurd Angenent
Department of Mathematics, University of Wisconsin – Madison
Daskalopoulos]Panagiota Daskalopoulos
Department of Mathematics, Columbia University, New York
Sesum]Natasa Sesum
Department of Mathematics, Rutgers University, New Jersey


P.
Daskalopoulos thanks the NSF for support in DMS-1266172.
N.
Sesum thanks the NSF for support in DMS-1056387.




  We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow
  that are rotationally and reflection symmetric.  We prove that these solutions are either the spheres or they all have unique asymptotic behavior as t→ -∞ and we give their precise asymptotic description.  This description applies in particular to the solution constructed by G.  Perelman in <cit.>.

[
    [
    December 30, 2023
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§ INTRODUCTION

A solution to a geometric evolution equation such as the Ricci flow, the Mean Curvature Flow or the Yamabe flow is called ancient if it exists for all t∈(-∞, t_0], for some t_0 ≤ +∞.  In the special case where the ancient solution
exists for all t ∈ (-∞, +∞), it is called eternal.
For all these flows, the requirement that a solution should exist for all times t≤ t_0, combined with some type of positive curvature condition or a suitable non-collapsedness assumption, turns out to be quite restrictive.  In a number of cases there are results which state that the list of possible ancient solutions to a given geometric flow consists of self similar solutions (“solitons”) and a shorter list of non self similar solutions.  Such classification
results play an important role in the singularity analysis of the flow, as blow up limits at a singularity give rise to an ancient solution.



Ancient compact solutions to the 2-dimensional Ricci flow were classified by Daskalopoulos, Hamilton and Sesum in <cit.>.
It turns out that in this case, the complete list contains (up to conformal invariance) only the shrinking sphere solitons and the King solution.
The latter is a well known example of ancient collapsed Ricci flow solution and is given by an explicit formula.
It was first discovered by J.  King <cit.> in the context of the logarithmic fast-diffusion equation on ^2 and later independently by P.  Rosenau <cit.> in the same context.
It also appears as the sausage model in the context of quantum field theory, in the independent work of Fateev-Onofri-Zamolodchikov <cit.>.
Although the King ancient solution is not a soliton, it may be visualized as two steady solitons, called “cigars”, coming from opposite spatial infinities, glued together.
Let us remark that the classification in <cit.> deals with both collapsed and non-collapsed solutions.
Non-compact ancient (eternal) solutions to the 2-dimensional Ricci flow were classified by Daskalopoulos and Sesum in <cit.> (see also in <cit.>).
It turns out that in this case the only eternal solutions with bounded curvature R(·,t) at each time slice, are the cigar solitons.  The results mentioned above classify all complete 2D-Ricci flow ancient solutions, both compact and non-compact.

Solutions analogous to the 2-dim Ricci flow King solution exist in the higher dimensional (n ≥ 3) Yamabe flow as well.  Again they are not solitons, although they are given in an explicit form
and were discovered by King <cit.> (and later independently by Brendle in a private communication with the authors).  However, in the case of the Yamabe flow many more solutions exist.  In <cit.>,
Daskalopoulos, del Pino, King and Sesum constructed a five parameter family of type I ancient solutions to the Yamabe flow on S^n, not given in closed form, however looking similar to the King solutions in <cit.>.
In fact the King solutions are part of this class of solutions that can be visualized as two shrinking solitons, coming from spatial infinities and glued together.
In addition in <cit.>, Daskalopoulos, del Pino, and Sesum constructed type II compact ancient solutions to the Yamabe flow on S^n (which they called a tower of bubbles).  Unlike the above mentioned type I ancient solutions,
the Ricci curvature of the tower of bubbles solutions changes its sign (they still have nonnegative scalar curvature).
The above examples show that the classification of closed ancient solutions to the Yamabe flow is very difficult, if not impossible.




For Curve Shortening (MCF for curves in the plane) Daskalopoulos, Hamilton, and Sesum <cit.> classified all ancient compact convex solutions by showing that the only possibilities are the shrinking circle and the paperclip solution.




The existence of higher dimensional ancient compact convex solutions was settled by White <cit.> and then also by Haslhofer and Hershkovits <cit.> who constructed solutions that are not solitons, and for which no closed form seems to exist.  In <cit.> formal asymptotics were derived for White and Haslhofer-Hershkovits' ancient solutions.  In <cit.> we showed that any ancient, rotationally symmetric, non-collapsed solution of MCF satisfies the unique asymptotics as t→-∞.  
The classification of such solutions was established in <cit.>,
where we show that every uniformly 2-convex ancient oval must be unique up to rotation, scaling, and translation and hence it must be the ancient oval solution constructed by White and by Haslhofer–Hershkovits (<cit.>).
This implies that every closed, uniformly 2-convex and non-collapsed ancient solution to the mean curvature flow must be either the family of contracting spheres or the unique, up to isometries, ancient oval.

The notion of non-collapsedness for mean convex Mean Curvature Flow was introduced by X.J. Wang (<cit.>) and subsequently was shown by Andrews (<cit.>) to be preserved along MCF.
It is known (<cit.>) that all non-collapsed ancient compact solutions to MCF are convex.
The non-collapsedness condition turns out to be important in the classification of ancient compact convex solutions, as evidenced by the “pancake type” examples which become collapsed as t→ -∞ (see <cit.> and <cit.> for the existence of such solutions and <cit.> for
a beautiful work on their classification under rotational symmetry).
On the other hand, ancient non compact non-collapsed uniformly 2-convex solutions were considered by Brendle and Choi in <cit.> and <cit.>, where the authors show that any noncollapsed, uniformly 2-convex non compact ancient solution to the mean curvature flow must be a rotationally symmetric translating soliton, and hence the Bowl soliton, up to scaling and isometries.



In this paper we focus our attention on 3-dimensional Ricci flow.  
Consider an ancient compact solution to the 3-dimensional Ricci flow

    ∂/∂ tg_ij = -2R_ij,

existing for t∈ (-∞,T) so that it shrinks to a round point at T.

Perelman <cit.> established the existence of a rotationally symmetric ancient κ-noncollapsed solution on S^3.  This ancient solution is of type II backward in time, in the sense that its scalar curvature satisfies

    sup_M× (-∞,0) |t|  |R(x,t)| = ∞

In forward time the ancient solution forms a type I singularity, as it shrinks to a round point.  Perelman's ancient solution has backward in time limits which are the Bryant soliton and the round cylinder S^2×ℝ, depending on how the sequence of points and times about which one rescales are chosen.  These are the only backward in time limits of Perelman's ancient solution.
Let us remark that in contrast to the collapsed  King ancient solution of the 2-dimensional Ricci flow, the Perelman ancient solution is
non-collapsed.  In fact there exist other ancient compact solutions to the 3-dimensional Ricci flow which are collapsed and the analogue of the King solution (see in <cit.>, <cit.>).  Let us recall the notion of κ-noncollapsed metrics,
which was introduced by Perelman in <cit.>.


  The metric g is called κ-noncollapsed on the scale ρ, if every metric ball B_r of radius r < ρ which satisfies
  | | ≤ r^-2 on B_r has volume at least κ  r^n.
  For any κ>0 an ancient Ricci flow solution is called κ-noncollapsed, if it is κ-noncollapsed on all scales ρ.


It turns out that this is an important notion in the context of ancient solutions and singularities.  In fact, in <cit.> Perelman proved that every ancient solution arising as a blow-up limit at a singularity of the Ricci flow on compact manifolds is κ-noncollapsed on all scales for some
κ >0.  In this context, the following conjecture plays an important role in the classification of singularities.



  
  Let (S^3,g(t)) be a compact ancient κ-noncollapsed solution to the Ricci flow (<ref>) on S^3.  Then g(t) is either a family of contracting spheres or Perelman's solution.


The Hamilton-Ivey pinching estimate tells us that any two or three dimensional ancient solution with bounded curvature at each time slice has nonnegative sectional curvature.  Since our solution (S^3,g(t)) is closed, the strong maximum principle implies
that the sectional curvatures, and hence the curvature operator, are all strictly positive.  Hence, Hamilton's Harnack estimate (<cit.>) implies that the scalar curvature of the solution g(t) has R_t ≥ 0, yielding that there exists a uniform constant C > 0 so that R(·,t) ≤ C, for all t∈ (-∞,t_0].  Since the curvature is positive, this yields the curvature bound

    _g(t)≤ C,         -∞ < t ≤ t_0,

and for some uniform constant C.

The previous discussion implies that any closed 3-dimensional κ-noncollapsed ancient solution is actually a κ-solution in the sense that was defined by Perelman in <cit.> (see Section <ref> for more details).

In a recent important paper <cit.>, Brendle proved that any three-dimensional non compact  ancient κ-solution is isometric to either a family of shrinking cylinders or their quotients, or to the Bryant soliton.  Brendle first shows that all three-dimensional ancient κ-solutions that are non compact must be rotationally symmetric.
He then shows that such a rotationally symmetric solution, if it is not a cylinder or its quotient, must be a steady Ricci soliton.
In the same paper Brendle states that his techniques can easily be applied to obtain the rotational symmetry of
three-dimensional compact ancient κ-solutions.  In fact this is shown in detail in a very recent work by Brendle <cit.>.
Also very recently Bamler and Kleiner in <cit.> obtained the same result in the compact case using different methods from Brendle.



The summary is that any solution satisfying the assumptions
of Conjecture <ref> is rotationally symmetric, hence reducing the resolution of Conjecture <ref> to establishing the classification of rotationally symmetric solutions.  The challenge in this problem comes from the fact that Perelman's solution is not given in explicit form and is not a soliton.  Similar challenge appears in the classification of ancient compact
non-collapsed MCF solutions which was resolved in <cit.>.
Because of the instability of the linearized operator at the cylinder,
which appears as asymptotic limit as t → -∞ in both flows, one needs to establish refined asymptotics for any ancient
solution under consideration as t → -∞.





In an attempt to resolve Conjecture <ref>, we will establish in this paper the (unique up to scaling) asymptotic behavior of any reflection and rotationally symmetric compact κ-noncollapsed ancient solution to the Ricci flow on S^3 which is not isometric to a round sphere.  Our main result states as follows.


  Let (S^3, g(t)) be any reflection and rotationally symmetric compact, κ-noncollapsed ancient solution to the Ricci flow on S^3 which is not isometric to a round sphere.
  Then the rescaled profile u(σ,τ) (solution to equation (<ref>)), defined on ℝ×ℝ, has the following asymptotic expansions:
  
    
  (i) For every L > 0,
    
    u(σ,τ) = √(2)(1 - σ^2 - 2/8|τ|) + o(|τ|^-1),     n   |σ| ≤ L

    as τ→ -∞.
    
  (ii) Define z := σ/√(|τ|) and u̅(σ,τ) := u(z√(|τ|), τ).
    Then,
    
    lim_τ→ -∞u̅(z,τ) = √(2 - z^2/2)

    uniformly on compact subsets of |z| < 2.
    
  (iii) Let k(t) := R(p_t,t) be the maximal scalar curvature which is attained at the two tips p_t, for t ≪ -1.  Then the rescaled Ricci flow solutions (S^3,g̅_t(s), p_t), with g̅_t(·, s) = k(t)   g(·,t+k(t)^-1  s), converge to the unique Bryant translating soliton with maximal scalar curvature one.
    Furthermore, k(t) and the diameter d(t) satisfy the asymptotics
    
    k(t) = log|t|/|t| (1 + o(1))      d(t) = 4√(|t|log |t|)  (1 +o(1))

    as t → -∞.
  


In a forthcoming work, we use Theorem <ref> to address Conjecture <ref>, in a similar way that results about unique asymptotics of ancient ovals shown in <cit.> were used to prove the classification result of closed ancient mean curvature flow solutions (see <cit.>).



As an immediate Corollary of the result in <cit.> and of Theorem <ref> we have the following result.


  Let (S^3, g(t)) be any reflection symmetric compact κ-noncollapsed ancient solution to the Ricci flow on S^3.  Then, it is rotationally symmetric and is either isometric to a round sphere or it has unique asymptotics which are given by Theorem <ref>.


In order to prove Theorem <ref> we will combine techniques developed in <cit.> and <cit.>.  In <cit.>, under the assumption on rotational symmetry, Brendle constructed barriers by using gradient Ricci solitons with singularity at the tip
which were found by Bryant (<cit.>).  In the proof of Theorem <ref> we use Brendle's barriers to localize our equation
in the parabolic region, similarly as in <cit.>.  The methods are similar to the ones used in <cit.>, but new difficulties arise due to the fact
that the equation (<ref>) is non-local.  The localization enables us to do spectral decomposition in the parabolic region and obtain refined asymptotics of our solution.



The outline of the paper is as follows: In section <ref> we discuss the backward in time limit of our solution and we list all equations under rotational symmetry, introducing different change of variables in different regions.  In section <ref> we use Brendle's barriers to achieve the spectral decomposition of our solution which yields precise asymptotics in the parabolic region.
Subsequently, we combine this exact behavior in the parabolic region together with barrier type arguments to obtain the precise behavior of our solution in the
intermediate region (see section <ref>).  In the last section
<ref>, we show the convergence of our solution to the Bryant soliton, after
appropriate rescaling and change of variables, and obtain the precise behavior of the
maximum scalar curvature and the diameter, as time approaches -∞.



§ PRELIMINARIES






 §.§ Equations under rotational symmetry



Assume that g is a solution of the Ricci flow (<ref>) on S^3, which is rotationally and reflection symmetric and shrinks to a round point at time T.  Since g(·,t) is rotationally symmetric, it can can be written as

    g = ϕ^2  dx^2 + ψ^2  g_ can,        (-1,1)× S^2

where (-1,1)× S^n may be naturally identified with the sphere S^3 with its North and South Poles removed.  The quantity ψ(x,t) > 0 is the radius of the hypersurface {x}× S^2 at time t.  By the reflection symmetry assumption we have ψ(x,t) =ψ(-x,t) for all x∈ (-1,1).
The distance function to the equator is given by

    s(x, t) = ∫_0^x ϕ(x', t)  dx'.

We will write

    s_±(t) := lim_x→± 1 s(x, t),

or shortly s_±, for the distance from the equator to the South and the North Poles, respectively.  Under Ricci flow the distances s_±(t) evolve with time.
If we abbreviate

    ds = ϕ(x, t)   dx      and     / s = 1/ϕ(x, t) / x

then we can write our metric as

    g = ds^2 + ψ^2  g_ can.

The time derivative does not commute with the s-derivative, and in general we
must use

    / tds = ϕ_t   dx = ϕ_t/ϕ  ds      and     [ / t, / s] =
    -ϕ_t /ϕ/ s.

The Ricci tensor is given by

    Rc    = 2K_0   ds^2 + [K_0 + K_1] ψ^2 g_ can
        = -2ψ_ss/ψ  ds^2
      + {
      -ψψ_ss - ψ_s ^2+1
      } g_ can

where K_0 and K_1 are the two distinguished sectional curvatures that any
metric of the form (<ref>) has.  They are the curvature of a plane tangent to {s}× S^n, given by

    K_1:= 1-ψ_s^2/ψ^2,

and the curvature of an orthogonal plane given by

    K_0 := -ψ_ss/ψ.

Moreover, the scalar curvature is given by

    R = g^jk R_jk
    = 4K_0 + 2K_1.

The time derivative of the metric is

    g/ t =
    2ϕ_t/ϕ  ds^2 + 2ψψ_t  g_ can.

Therefore, since the metric g(t) evolves by Ricci flow (<ref>), we have

    ϕ_t/ϕ = 2ψ_ss/ψ,

so that

    t   ds = 2ψ_ss/ψ  ds      and     [ t,  s]
    = -2ψ_ss/ψ  s.

Under Ricci flow the radius ψ(s,t) satisfies the equation

    ψ_t = ψ_ss - 1-ψ_s^2/ψ.

As in <cit.>, for our metric (<ref>) to define a smooth metric on S^3 we need to have

    ψ_s(s_-) = 1,     ψ^(2k)(s_-) = 0         ψ_s(s_+) = -1,     ψ^(2k)(s_+) = 0,

for k∈ℕ∪{0}.



Consider next the rescaled function

    u(x, τ) := ψ(x,t)/√(T-t),

as well as the rescaled time and distance to the equator

    σ(x,t) := s(x,t)/√(T-t),   τ = -log (T-t).

If we write

    dσ = ds/√(T-t),     /σ = √(T-t) / s,    τ = (T-t) t,

then we get

    τ dσ = (1/2 + 2u_σσ/u)dσ.

For the commutator we get

    [τ, σ]
      = - (1/2 + 2u_σσ/u) σ.

We write σ_±(t) = s_±(t)/√(T-t), or simply σ_± for the rescaled distance from the equator to the poles.
The rescaled radius u:(-1,1)×(-∞, 0)→ satisfies the equation

    u_τ = u_σσ + u_σ^2/u - 1/u + u/2

with boundary conditions

    u_σ = ∓ 1,  
      u^(2k)=0      at σ=σ_±(τ),   τ<0.





  §.§.§ The expansion term

By analogy with similar equations in MCF one might expect a term of the form -1/2σ u_σ in (<ref>).  However, in the present set-up that term is replaced by a change in the commutator [_τ, _σ] (in (<ref>) or (<ref>)) which accounts for the stretching involved in passing from the s to σ coordinate.



The vector fields ∂_τ and ∂_σ do not commute.  To overcome this we replace the vector field _τ with

    D_τ = _τ - I  _σ

for some function I, and we require that D_τ and _σ commute.  It follows from

    [D_τ, _σ]
    = [_τ, _σ] + I_σ_σ
    = {-1/2 - 2u_σσ/u + I_σ}_σ.

Hence D_τ and _σ commute if

    I(σ, t) = 1/2σ + J(σ, t),    
      J(σ,t) := 2 ∫_0^σu_σσ/u  dσ'.


We can then write the equation (<ref>) for u as

    D_τ u = u_σσ - σ/2  u_σ
      - J(σ,τ)   u_σ + u_σ^2/u - 1/u + u/2.

One can think of D_τ as “the derivative with respect to τ keeping σ constant,” while _τ is “the τ-derivative keeping x fixed.”
We will abuse notation and write u_τ both for _τ u and for D_τ u, when it is clear from the context which time derivative we mean.
For instance, we will write equation (<ref>) as

    u_τ = u_σσ - σ/2  u_σ
      - J(σ,τ)   u_σ + u_σ^2/u - 1/u + u/2.



  The solution u ≡√(2) corresponds to the shrinking cylinder soliton S^2×ℝ.


The representation g=ϕ^2   dx^2+ψ^2  g_ can leads to singularities at the poles x=± 1.  We overcome this difficulty by choosing new local coordinates.  As in Angenent-Caputo-Knopf <cit.>, we regard ψ(s,t) as a new local coordinate on any interval (x_0, x_1) on which ψ(x, t) is a monotone function of x.
By our assumptions ψ(x,t) is a concave function of s (i.e. ψ_ss<0).  This implies ψ(x, t) is strictly increasing for x<0 and strictly decreasing for x>0.  In the region x<0 (the southern hemisphere) we take ψ as coordinate, and express the metric and all its components as functions of (ψ,t).  The metric is then

    g = ϕ^2  dx^2 + ψ^2 g_ can,

where we still have to write (ϕ  dx)^2 in terms of ψ.  We have

    ϕ(x, t)   dx = ds = dψ/ψ_s

so that

    g = dψ^2/ψ_s^2 + ψ^2  g_ can.

To describe the evolution of the metric we must therefore keep track of the quantity Y̅ := ψ_s^2 as a function of (ψ, t).  That is, we set

    Y̅(ψ, t) = ψ_s^2(s,t),     ψ=ψ(s,t).

A direct calculation shows that it evolves by the PDE

    Y̅_t =
      Y̅Y̅_ψψ
      - 1/2  (Y̅_ψ)^2
      + (1 - Y̅) Y̅_ψ/ψ
      + 2 (1-Y̅)  Y̅/ψ^2.

In this equation _t stands for the derivative with respect to time at constant ψ.  Therefore _t and _ψ commute.

As above, we will also work here with the rescaled variables (u,τ) given by (<ref>).  Thus we introduce Y:=u_σ^2
as a function of (u,τ), that is

    Y̅(ψ, t) = Y(u, τ),      u=ψ/√(T-t), τ = -log(T-t).

A short computation then shows that Y(u,τ) evolves by

    Y_τ + u/2  Y_u = Y Y_uu - 1/2  (Y_u)^2 + (1 - Y) Y_u/u + 2 (1-Y)  Y/u^2.






 §.§ Backward limit of our solution


We will first analyze the backward limit of any κ-solution.  Let t̅ = -t.  By the work of Perelman (<cit.>, see also <cit.> for details), for every t̅ > 0 there is some q(t̅) ∈ M so that l(q(t̅),t̅) ≤ 3/2, where
l(q,t̅) denotes the reduced distance

    l(q,t̅) := 1/2√(t̅)inf_γ∫_0^t̅√(t̅') (R(γ(t̅')) + |γ̇|^2)  dt̅',

and where R(γ(t̅'), t̅') and |γ̇| are computed using the metric at -t̅' and γ is any curve connecting some fixed point p and any point q so that γ(0) = p and γ(t̅) = q.  In <cit.> Perelman showed that for a subsequence of t̅_i := -t_i (call it again t̅_i), the parabolically rescaled sequence of metrics g_i(t), around l(q(t̅_i),t̅_i), t_i), converges to a non flat gradient shrinking Ricci soliton.  By the classification result of such solitons we know they are either the spheres or round cylinders S^2×ℝ.  The limiting gradient soliton is called an asymptotic soliton.  


  
  Assume (S^3,g(t)) is a closed κ-solution whose asymptotic soliton is a round sphere.  Then the Ricci flow solution g(·,t), t ∈
  (-∞,T) must itself be a family of shrinking round spheres.



  By our assumption, (S^3, g_i(t)), where g_i(·,t) = 1/(-t_i)  g(t_i - t   t_i), converges to a sphere as i→∞.  Then there exist a ρ > 0 and an i_0 > 0, so that for all i≥ i_0 we have (g_i(0)) ≥ρ  R(g_i(0))  g_i(0), or equivalently, (g(t_i)) ≥ρ R(g(t_i))  g(t_i).  By Theorem 9.6 in <cit.> we have that (g(t)) ≥ρ  R(g(t))  g(t) holds for all t ≥ t_i.  Since t_i→ -∞, we get that
  
    (g(t)) ≥ρ R(g(t))  g(t),        t ∈ (-∞,T).

By the result in <cit.>, this yields that (S^3,g(t)) has constant positive sectional curvature, for all t ∈ (-∞,T), and hence our solution is a family of shrinking spheres.  



Let us now assume that (S^3, g) is a κ-solution which is rotationally and reflection symmetric.  We have seen in
the previous subsection that g may be expressed in the form (<ref>) and that because of reflection symmetry x = 0 can be taken to correspond to a point of maximum radius ψ for every t ≤ t_0.  We claim the following.


  
  There exist uniform constants C < ∞ and t_0 ≪ -1 so that
  
    R(q,t) ≤C/|t|,         t ≤ t_0

  where q∈ M corresponds to x = 0 and hence s = 0, for all t ≤ t_0.



  By the maximum principle applied to ψ that satisfies (<ref>) we have
  
    d/dtψ_max≤ - 1/ψ_max.

  Integrating this differential inequality on [t, t_1], for a fixed t_1, yields that ψ_max^2(t) ≥ 2|t| + ψ^2_max(t_1) - 2 |t̅_1| ≥ |t|,
  for all t ≤ t_0 ≪ -1.  This implies the bound
  
    K_1(q,t) = 1 -ψ_s^2/ψ^2≤1/|t|,        t ≤ t_0 ≪ -1.

  
  Next, we claim there exist uniform constants C and t_0 ≪ -1 so that
  
    K_0(q,t) ≤ C  K_1(q,t),        t ≤ t_0 ≪ -1.

  To prove this claim we argue by
  contradiction.  Assume that the claim is not true, meaning there exist a sequence
  t_i → -∞ and C_i →∞ so that 
    K_0(q,t_i) ≥ C_i K_1(q,t_i).

  We rescale the Ricci flow solution g(·,t) by R(0,t_i) around (q,t_i), where
  q∈ M corresponds to s = 0.  By Perelman's compactness theorem for
  κ-solutions, there exists a subsequence of rescaled solutions converging to
  another rotationally symmetric κ-solution, which, in view of
  (<ref>), implies that on the limiting solution one has
  K^∞_1 ≡ 0 and K^∞_0≢0 on the limit
  M_∞.  Since our limiting rotationally symmetric metric g_∞ is of
  the form g_∞ = ds_∞^2 + ψ_∞(s)^2 g_ can, where
  K^∞_1 = 1 - ψ_∞ s^2/ψ_∞^2≡ 0, we get
  ψ^2_∞ s≡ 1.  This implies that
  ψ_∞,ss≡ 0, which contradicts K^∞_0≢0.
  We conclude that (<ref>) holds which combined with (<ref>) finishes the proof of the Lemma.


By the previous discussion in this section (in particular Lemma <ref>), we may assume in the rest of the paper that the asymptotic soliton of our κ-solution is a round cylinder.
Define the parabolically rescaled metric

    g̃(·,τ) := 1/(-t)  g(·,t),     τ = -log(T - t).

Let q ∈ S^3 be the point as in Lemma <ref>.  Then we have the following result.


  
  Let (S^3,g(·,t)) be a closed κ-solution whose asymptotic soliton is a round cylinder.  Then the rescaled solution
  (S^3, g̃(·,τ)) around a fixed point q converges in Cheeger-Gromov sense to the round cylinder of radius √(2).



  See Proposition 3.1 in <cit.>.




 §.§ Radially symmetric barriers from <cit.>


S.  Brendle in <cit.> constructed barriers for equation (<ref>).  His construction gives the following result which will be used in the next sections.


  There exists a one parameter family Y_a:=Y_a(u) of super-solutions to the elliptic equation
  
    Y_a Y_a” - u/2  Y_a' - 1/2  (Y_a')^2 + (1 - Y) Y_a'/u + 2 (1-Y_a)  Y_a/u^2 <0

  which are defined on u ∈ [r^* a^-1, 9/8√(2) ], for a fixed number r^* >0.  Moreover, there exists a
  small constant η >0 which is independent of a and a smooth function ζ(s) with 2+ζ(√(2)) =1/4
  such that
  
    Y_a(u) = a^-2  (2 u^-2 -1) + a^-4  (8 u^-4 + ζ(u) ) + O(a^-6)

  for all u ∈ [√(2)-η, √(2)+η] and a ≫ 1.
  


Note that Brendle in <cit.> considers solutions of

    Y_a Y_a” - u/2  Y_a' - 1/2  (Y_a')^2 + (1 - Y_a) Y_a'/u + 2 (1-Y_a)  Y_a/u^2 <0

on u ∈ [r^* a^-1, 9/8 ] since he scales u= ψ/√(-2t), but we will consider solutions of (<ref>) on
u ∈ [a^-1  r^*, 9/8√(2) ], for a different constant r^*, to be consistent with our scaling u= ψ/√(-t).
We refer the reader to <cit.> Section 2 (Propositions 2.4 and 2.5) for the details of the construction of the barriers Y_a
and their properties.



We next state the following result, which is Proposition 2.8 in <cit.> adopted to our notation.


  There exists a large number K with the following property.  Suppose that a ≥ K and τ̅≪ -1.  Moreover, suppose that u̅(τ) is a function satisfying | u̅(τ) - √(2)| ≤1/100  a^-2 and Y(u̅(τ),t) ≤1/32  a^-4 for all τ≤τ̅.  Then
  
    Y(u,τ ) ≤ Y_a(u),        r_* a^-1≤ u ≤u̅(τ),    τ≤τ̅.

  In particular, Y(u,τ) ≤ C  a^-2 for all 1/2≤ u ≤u̅(τ) and τ≤τ̅.


The above proposition will play crucial role in Section <ref> as it will allow us to introduce cut-off functions supported in the parabolic region of our solution.  For this application we will
need to have the result of the Proposition <ref> holding up to the maximum point of our solution u(σ,τ) of (<ref>).  Since we have assumed reflection symmetry this corresponds to σ=0.
However at the maximum u(0,τ) we have Y:=u_σ^2=0 and equation (<ref>) becomes degenerate.  In the following consequence of Proposition <ref>, we justify that the comparison principle
can be extended up to Y=0 and allow us to have the result in this proposition with u(τ):= u(0,τ) = max u(·,τ).




  There exists a large number K with the following property.  Suppose that a ≥ K and τ̅≪ -1.
  Moreover, suppose that | u(0, τ) - √(2)| ≤1/200  a^-2 for all τ≤τ̅.  Then
  
    Y(u,τ ) ≤ Y_a(u),        r_* a^-1≤ u ≤ u(0,τ),    τ≤τ̅.

  In particular, Y(u,τ) ≤ C  a^-2 for 1/2≤ u ≤ u(0,τ) and all τ≤τ̅.


 We will apply Proposition <ref> for u(σ,τ), with |σ| ≤δ≪ 1, and then let σ→ 0.  Since we have assumed reflection symmetry it is sufficient to assume that σ >0.
  Let a >0 be such that | u(0, τ) - √(2)| ≤1/200  a^-2 for all τ≤τ̅.  The uniform continuity
  of both u (·, τ) and u_σ(·,τ) on |σ | ≤ 1, τ≤τ̅ (this follows from the smooth uniform convergence on compact sets
  lim_τ→ -∞ u (·, τ) = √(2)) combined with u_σ(0,τ)=0 imply that there exists δ(τ̅, a) such that
  
    | u(σ, τ) - √(2)| ≤1/100  a^-2         |u_σ^2(σ,τ)| ≤1/32 a^-4

  for all |σ| ≤δ(τ̅, a), τ≤τ̅.  It follows that the assumptions of Proposition <ref>
  are satisfied for u(σ,τ) (recall that Y(u(σ, τ),τ) = u_σ^2(σ,τ)).  Therefore, Proposition <ref>
  yields that for all 0 < σ < δ(τ̅, a),
  
    Y(u,τ ) ≤ Y_a(u),        r_* a^-1≤ u ≤ u(σ,τ),    τ≤τ̅.

  Letting σ→ 0 we conclude the desired result.






§ PARABOLIC REGION ASYMPTOTICS




Our goal in the next two sections is to establish the asymptotic behavior of any
rotationally symmetric solution u(σ,τ) of (<ref>) (or
equivalently (<ref>)) in the cylindrical region

    _θ := { σ : |    u(σ, τ) ≥θ/2  }

for any θ >0 small.  This will be done in two steps: in this section, we will first
analyze the linearized operator or our equation at the cylinder u=√(2) to
establish precise asymptotics of u(σ,τ) in the parabolic region
_L := { σ :    |σ | ≤ L  }, holding for any
L ≫ 1 and then, in the next section, we will use these asymptotics and barrier arguments to
establish the behavior of u(σ,τ) in the intermediate region
_L,θ := { σ :   |σ| ≥ L,    u(σ,τ)
≥ θ/2  }. Because of reflection symmetry it is enough to consider only the case where σ≥ 0.



We will assume throughout this section that u(σ,τ) is a solution of
(<ref>) (in commuting variables).  The cylindrical norm in the parabolic region is defined as

    sup_τ' ≤τf(·,τ'),

where f: × (-∞,τ_0] → and

    f(·,s)^2 
    := ∫_ f(σ,s)^2   e^-σ^2/4  dσ.

We will often denote dμ := e^-σ^2/4  dσ.  We will prove the following:

 
  Under the same assumptions as in Theorem <ref>, for any L ≫ 1, the solution u(σ,τ) of (<ref>) satisfies the following asymptotics
  
    u(σ,τ) 
        = √(2) ( 1 - σ^2-2/8 |τ| 
        + o (1/|τ|) )
             (τ→-∞)

  uniformly for |σ| ≤ L.


To prove Theorem <ref> we will use crucial ideas from a
recent paper by Brendle <cit.> and combine them with our methods in <cit.>.  In
what follows below we will outline those crucial ideas from <cit.>, and for
the proofs of those we refer the reader to the same paper.



To prove the asymptotics
estimates in <cit.> in the compact case and in <cit.> and <cit.> in the
non-compact case, the spectral decomposition of the linearized operator at the cylinder in terms of
Hermite polynomials has been used.  The localization argument used in both papers, <cit.> and <cit.>, to make the spectral decomposition possible uses a calibration argument which has no obvious analogue in the Ricci flow.  The main obstruction comes from the non-local character of (<ref>),
as the construction of barriers plays an important role.
In <cit.>, Brendle manages

to use barrier arguments.  For the barrier construction he uses steady gradient Ricci
solitons with singularity at the tip, which were found by Bryant in <cit.>.
The same technique plays an important role in our case.

In order to set up the barrier argument we will change the variables and
consider our evolving metrics written in the form
g̃ = Y(u,τ)^-1 (du)^2 + u^2 g_S^2, where
Y(u,τ) = u_σ^2.  Note also that u_σ = ψ_s, so will
sometimes consider Y as a function of ψ.  Since u_σ = 0 at
σ = 0, this change of variables is good from u = 0 all the way up to
u = u(0,τ), but not including u = u(0,τ).



 §.§ Analysis near the cylinder soliton

Let v(σ,τ) be such that

    u(σ,τ) = √(2) (1 + v(σ,τ)).

The function v satisfies

    v_τ = v_σσ
      -(σ/2+J[v]) v_σ
      +v
      +v_σ^2/1+v
      - v^2/2(1+v),

where the function J[v] is defined by

    J[v](x, t) = 2∫_0^x v_σσ/1+v  dσ
      =2∫_0^x dv_σ/1+v.

We can split the terms in (<ref>) into linear and higher order terms as follows

    v_τ =  [v] +  [v]

where

    [v] := v_σσ -σ/2 v_σ + v

and

    [v] := -J[v]  v_σ +v_σ^2/1+v - v^2/2(1+v).




The function v(σ,τ) is not defined for all σ∈ and therefore does not belong to the Hilbert space
= L^2( , e^-σ^2/4  dσ) at any time τ.  Since this is the natural Hilbert space to consider, we truncate v outside an interval |σ|≥σ_*(τ) for a suitably chosen σ_*(τ), which we will allow to depend on the size of v(σ,τ) at some fixed finite value of σ.
Thus we follow Brendle <cit.> and define for each τ

    δ(τ)
         := sup_τ'≤τ (
      |u(0, τ')-√(2)| + |u_σ(0,τ')|
       ) = √(2) sup_τ'≤τ
      |v(0, τ')|

since u_σ(0,τ) = 0 for all τ.


By Lemma 3.8 in <cit.> we have that there exists a uniform constant C so that for τ≤τ_0 ≪ -1 we have

    χ_[-δ^-θ, δ^-θ] (v_τ - ℒ v)^2 ≤ C δ^θ χ_[-1/2δ^-θ, 1/2δ^-θ] v^2 + C  e^-1/8 δ^-2θ,

where χ_[-δ^-θ, δ^-θ] and χ_[-1/2δ^-θ, 1/2δ^-θ] are the characteristic functions of the sets [-δ^-θ, δ^-θ] and [-1/2δ^-θ, 1/2δ^-θ], respectively, and θ = 1/100.  Note that we have suppressed the dependence of δ on τ and we simply wrote δ for δ(τ).  We will do that below as well when there is no ambiguity.

Choose a cutoff function χ̂∈ C^∞() satisfying χ̂= 1 on [-1/2,1/2] and χ̂= 0 outside of [-1,1], and set

    χ(σ,τ) = χ̂(δ(τ)^θσ),    θ=1/100.

We can now introduce our truncated version of v:

    v̅(σ,τ) := v(σ,τ) χ(σ, τ).

It easily follows that

    v̅_τ - ℒv̅ = χ (v_τ - ℒ v) + (χ_τ - χ_σσ + σ/2χ_σ)  v - 2χ_σ v_σ =: E_0(σ,τ).


We claim that there exists a uniform constant C so that for all τ≤τ_0 ≪ -1 we have

    E_0(·,τ)^2 ≤ C δ^θv̅^2 + C  e^-1/8 δ^-2θ.

Indeed, by (<ref>) we immediately get that

    χ (v_τ - ℒ v)^2 ≤ C δ^θ v̅^2 + C   e^-1/8 δ^-2θ.

By the definition of χ in (<ref>) we have

    (χ_τ - χ_σσ + σ/2 χ_σ)  v - 2χ_σ v_σ^2 ≤ C  e^-1/8 δ^-2θ.

Combining those two estimates immediately yields (<ref>).













  The function δ(τ) is non-decreasing with lim_τ→ -∞δ(τ) = 0.  Moreover, δ(τ) is Lipschitz continuous; in particular, δ(τ) is absolutely continuous and its derivative δ'(τ) is uniformly bounded and non negative a.e.


  The definition of δ(τ) directly implies that δ(τ) is non-decreasing.
  As τ→-∞ we have u(σ,τ)→√(2) uniformly for |σ|≤ 2σ_0.
  By parabolic regularity it follows that all derivatives of u are uniformly bounded for |σ|≤σ_0 and for τ≪ 0.
  Hence u(0, τ) is a smooth function of τ with uniformly bounded time derivatives.
  This implies that δ(τ) is a Lipschitz function of τ.
  Rademacher's Theorem implies that δ(τ) is absolutely continuous and, in particular, differentiable almost everywhere.


We have the following crucial lemma which will allow us to control error terms coming from cutoff functions.

   There exist uniform constants τ_0 ≪ -1 and C > 0 so that
  
    |v_σ(σ,τ)| + |v(σ,τ)| ≤ C  δ(τ)^1/8,

  for |σ| ≤δ(τ)^-θ.  Moreover,
  
    δ(τ)^4
      =  ( sup_τ' ≤τ |v(0,τ')| )^4
      ≤ C sup_τ'≤τv̅(τ')^2.




  The first estimate follows by the same proof as of Lemma 3.7 in <cit.>.  The second estimate is also shown in <cit.> and follows by standard interpolation inequalities.


We split the proof of Theorem <ref> in a few propositions.  First, we derive a more detailed equation for v̅.


  The function v̅ satisfies the linear inhomogeneous equation
  
    v̅_τ - v̅
        = a(σ,τ)v̅_σ +
        b(σ,τ)v̅ + c(σ,τ),

  where
  
    a(σ,τ) = -J[v] +v_σ/1+v,     b(σ,τ) = - v/2(1+v),

  and
  
    c(σ,τ) =
        {
        - v v_σ/1+v
        +J[v]v
        -2 v_σ}χ_σ
        +(χ_τ - χ_σσ + σ/2χ_σ) v   .



  In (<ref>) we have an evolution equation for v̅.
  Using (<ref>) and also χ v_σ=v̅_σ - χ_σ v, we then get
  
    v̅_τ - v̅   = χ· (v_τ- v) + (χ_τ - χ_σσ+σ/2χ_σ) v
        -2χ_σ v_σ
       = χ v_σ^2/1+v
        -χ v^2/2(1+v) -J[v]  χ v_σ
        +(χ_τ - χ_σσ + σ/2χ_σ) v
        -2χ_σ v_σ
       = (-J[v] +v_σ/1+v) v̅_σ
        -v/2(1+v)v̅
           
        -χ_σ v v_σ/1+v
        +J[v]  χ_σ v
        +(χ_τ - χ_σσ + σ/2χ_σ) v
        -2χ_σ v_σ.

  


We can apply the variation of constants formula to (<ref>), which tells us that for any given τ≪ 0 one has

    v̅ (τ) = e^v̅(τ-1)
      + ∫_τ-1^τ e^(τ-τ'){
      a  v̅_σ(σ, τ') + b  v̅(σ, τ')
      + c(σ, τ')
      }
      dτ'.

Here e^t is the heat semigroup on the Hilbert space = L^2(, dμ) associated with the operator .  It has the usual smoothing property (<ref>), i.e.    it satisfies (1-)^re^t≤ Ct^-re^t for all t>0.



To use (<ref>) we first estimate the coefficients a, b, and c.


  Given any ϵ>0 there is a τ_0≪ 0 such that for all τ≤τ_0 and |σ| ≤δ(τ)^-θ one has
  
    |a(σ, τ)| + |b(σ,τ)| ≤ C  ϵ .

  One also has
  
    c(·, τ)≤ C   e^-1/64δ(τ)^-2θ.

  The derivatives of a, b, and c satisfy the following estimates
  
    |a_σ|≤ C,      |b_σ|≤ Cϵ,    and   c_σ(·, τ)≤ C e^-1/64δ(τ)^-2θ.



  Given any ϵ>0, Lemma <ref> guarantees the existence of a τ_0≪0 such that for all τ≤τ_0 and |σ|≤δ(τ) one has |v(σ,τ)|+|v_σ(σ,τ)|≤ϵ.
  For the nonlocal term J[v] we therefore have
  
    |J[v]| ≤∫_0^σ|dv_σ|/1+v≤ Cϵ.

  The coefficient c(σ,τ) is bounded by
  
    |c(σ,τ)| ≤ C   |χ_σ| + |χ_τ - χ_σσ + 1/2σχ_σ|

  Recall that χ(σ,τ)=χ̂(δ(τ)^θσ).  This implies that
  
    |χ_σ| ≤ Cδ(τ)^θ,      
      |χ_σσ| ≤ C δ(τ)^2θ,      
      |σχ_σ| ≤ C,

  and also
  
    |χ_τ(σ,τ)| ≤θ δ(τ)^θ-1δ'(τ)  σ χ'(δ(τ)^θσ) ≤ C  δ(τ)^-1 δ'(τ).

  The hardest term to estimate is χ_τ (·, τ).  We have
  
    χ_τ (·, τ)^2 ≤ C∫_1/2δ(τ)^-θ^∞δ(τ)^-2δ'(τ)^2 e^-σ^2/4 dσ≤ C δ(τ)^-1δ'(τ)^2 e^-1/16δ(τ)^-2θ

  which implies
  
    χ_τ (·, τ)≤ C  δ(τ)^-1/2 e^-1/32δ(τ)^-2θδ'(τ).

  The desired estimate for χ_τ (·, τ) now follows from Lemma <ref> which guarantees that δ'(τ) is uniformly bounded.  The rest of the terms in c(σ, τ) are easy to estimate.
  
  To estimate the derivatives of the coefficients we use Lemma <ref> which tells us that
  
    0≤ -u_σσ≤1-u_σ^2/u≤1/u,

  and hence that u_σσ and v̅_σσ are uniformly bounded for |σ| ≤ 2δ(τ)^-θ.
  We have
  
    a_σ = 2u_σσ/u + v_σσ/1+v -v_σ^2/(1+v)^2,

  which implies that a_σ is uniformly bounded if |σ| ≤ 2δ(τ)^-θ.  For b_σ we have
  
    b_σ = v_σ/2(1+v)^2,

  which implies |b_σ|≤ Cϵ.  Finally, for c_σ we differentiate (<ref>) with respect to σ, which leads to many terms, namely
  
    c/σ= {(n-1) v v_σ/1+v -J[v]v -2 v_σ}χ_σσ + χ_σ/σ{(n-1) v v_σ/1+v -J[v]v -2 v_σ}
    
        +v  /σ(χ_τ - χ_σσ + σ/2χ_σ) +(χ_τ - χ_σσ + σ/2χ_σ) v_σ .

  Using χ(σ,τ) = χ̂(δ(τ)^θσ), the boundedness of δ'(τ) (Lemma <ref>), and our bounds for v, v_σ, and v_σσ we find that c_σ is bounded by C  δ(τ)^m,
  for some constants C and m.  Furthermore c_σ is supported in the region where 1/2δ(τ)^-θ≤σ≤δ(τ)^-θ.  This implies the stated estimate for c_σ.
  




 §.§ Spectral decomposition and the dominant mode

Our linearized operator  given by (<ref>) is self-adjoint in the Hilbert Space

    = L^2( , e^-σ^2/4  dσ),     ⟨ f, g⟩
    = ∫_ f(σ) g(σ) e^-σ^2/4  dσ.

It satisfies

    1- = _σ^*_σ,

where _σ^*=-_σ+σ/2 is the adjoint of _σ.  Using the identity [_σ^*, _σ] = -1/2 one finds that

  
  
    f_σ^2 = ⟨ f, (1-)f⟩ = √(1-)f^2 
       f_σσ^2 + 1/2f_σ^2
        = (1-)f^2





The operator  generates an analytic semigroup on  and by the spectral theorem one has the estimates

    (1-)^re^tf≤C_r/t^re^t

for all t>0.

The operator  has a discrete spectrum.  When restricted to reflection symmetric functions, its eigenvalues
are given by {λ_k}_k=0^∞, where λ_k = 1 - k.  The corresponding eigenvectors are the Hermite polynomials h_2k(σ), where h_0(σ) = 1, h_2(σ) = σ^2 - 2, etc.  Let us write = _0 ⊕_+ ⊕_-, where _+ is spanned by h_0, _0 is spanned by h_2, and _- is spanned by the remaining eigenfunctions {h_4, h_6, …},
and denote by 𝒫_+, 𝒫_0 and 𝒫_- denote the orthogonal projections associated with the direct sum = _0 ⊕_+ ⊕_-.  Also, we define

    _±(·, τ) = 𝒫_±[(·, τ)],  _0(·, τ) = 𝒫_0[(·, τ)]

so that

    (y,τ) = _+(y,τ) + _0(y,τ) + _-(y,τ).

It will be convenient to abbreviate

    Γ(τ) := sup_τ'≤τ v̅(·,τ')

and similarly

    Γ_+(τ) := sup_τ'≤τv̅_+(τ'),     Γ_-(τ) := sup_τ'≤τv̅_-(τ'),     Γ_0(τ) := sup_τ'≤τv̅_0(τ').




In Proposition <ref> we will show that |Γ_±(τ)| are small compared to the quantity

    α_*(τ) = sup_τ'≤τ |α(τ)|,     α(τ) = ∫v̅  h_2   e^-σ^2/4  dσ.

and show that the latter dominates for τ≪ -1.  Note that by definition α_*(τ) is a nondecreasing function of τ and that
Γ_0(τ) = α_*(τ)  h_2 but it is more convenient
in terms of notation to work with α_*(τ). The first step to showing this result is the next lemma
where we show that
either Γ_+(τ) dominates or Γ_0(τ) does.

 Either
  
    Γ_-(τ) + Γ_0(τ) = o(Γ_+(τ))
           (τ≤τ_0),

  or
  
    Γ_-(τ) + Γ_+(τ) = o(Γ_0(τ))
           (τ≤τ_0).




  Using the variation of constants formula we can represent the solution v̅(σ,τ) to (<ref>) as
  
    v̅(τ+1) = e^ℒv̅(τ) + ∫_τ^τ+1 e^(τ+1-τ') ℒ  E_0(·,τ')  dτ'.

  Applying the projections 𝒫_±, and using the description of the spectrum of ℒ given above, we have the following representations for the projections _± and _0:
  
    v̅_+(τ+1) = e v̅_+(τ) + ∫_τ^τ+1 e^τ+1-τ' 𝒫_+E_0(·,τ')  dτ' 
       v̅_-(τ+1) = e^ℒv̅_-(τ) + ∫_τ^τ+1 e^(τ+1-τ') ℒ 𝒫_-  
        E_0(·,τ')  dτ'
       v̅_0(τ+1) = v̅_0(τ) + ∫_τ^τ+1𝒫_0  E_0(·,τ')   dτ'.

  By (<ref>) we have
  
    v̅_+(τ+1)≥ e v̅_+(τ) - C ∫_τ^τ+1δ(τ')^θ/2 v̅(τ') dτ' - C ∫_τ^τ+1 e^-1/16δ(τ')^-2θ dτ'.

  Given τ_1 < 0, choose τ_2 ≤τ_1 so that Γ_+(τ_1) = v̅(τ_2).  Then, using that both Γ_+(τ) and δ(τ) are nondecreasing functions in time and the last estimate, we obtain
  
    Γ_+(τ_1+1)    ≥v̅_+(τ_2+1)
       ≥ e  v̅_+(τ_2) - C ∫_τ_2^τ_2+1δ(τ')^θ/2 v̅(τ') dτ' - C ∫_τ_2^τ_2+1 e^-1/16δ(τ')^-2θ dτ' 
       ≥ e Γ_+(τ_1) - C δ(τ_2 + 1)^θ Γ_+(τ_2 + 1) - C  e^-1/16δ(τ_2+1)^-2θ
       ≥ e Γ_+(τ_1) - C δ(τ_1 + 1)^θ Γ_+(τ_1 + 1) - C  e^-1/16δ(τ_1+1)^-2θ.

  This implies that for every τ≤τ_0 ≪ -1 we have
  
    Γ_+(τ) ≤ e^-1 Γ_+(τ+1) + C δ(τ + 1)^θ Γ_+(τ + 1) + C  e^-1/16δ(τ+1)^-2θ.

  Regarding the negative mode, using (<ref>) and the representation for v̅_- we also have that
  
    v̅_-(τ+1)≤ e^-1 v̅_-(τ) + C ∫_τ^τ+1δ(τ')^θ/2 v̅(τ')  dτ' + ∫_τ^τ+1 e^-1/16δ(τ')^-2θ  dτ',

  implying that for all τ_1 ≤τ_0, we have
  
    Γ_-(τ_1 + 1)    := sup_τ≤τ_1v̅_-(τ+1)
       ≤ e^-1 sup_τ≤τ_1v̅_-(τ) + C δ(τ_1+1)^θ/2Γ_-(τ_1 + 1) + C  e^-1/16δ(τ_1+1)^-2θ.

  Since τ_1 ≤τ_0 ≪ -1 is arbitrary, we immediately get that for all τ≤τ_0 ≪ -1 the following holds
  
    Γ_-(τ) ≥ e Γ_-(τ+1) - C δ(τ+1)^θ/2Γ_-(τ + 1) - C  e^-1/16δ(τ+1)^-2θ.

  Finally, using the representation for v̅_0 and (<ref>), similarly to the other two projections we obtain
  
    |Γ_0(τ+1) - Γ_0(τ)| ≤ C δ(τ+1)^θ/2Γ_0(τ + 1) + C  e^-1/16δ(τ+1)^-2θ.

  
  Next, the key observation (as in <cit.>) is that
  
    δ(τ)^4 ≤ C  Γ(τ),

  by standard interpolation inequalities.  Consequently,
  
    e^-1/16δ(τ+1)^-2θ≤ C δ(τ)^5 ≤ C δ(τ) Γ(τ).

  Thus, we conclude that
  
    Γ_+(τ) ≤ e^-1Γ_+(τ+1) + C δ(τ+1)^θ/2 Γ_+(τ+1) 
        |Γ_0(τ+1) - Γ_0(τ)| ≤ C δ(τ+1)^θ/2 Γ_0(τ+1)
       Γ_-(τ) ≥ e Γ_-(τ+1) - C δ(τ+1)^θ/2 Γ_-(τ+1).

  The rest of the proof is identical to the proof of Lemma 3.9 in <cit.>.


We will next show the crucial for our purposes result which states that Γ_0(τ) actually dominates for τ≪ -1.

 
  There exists a τ_0 ≪ -1 so that for all τ≤τ_0 we have
  
    Γ_+(τ) + Γ_-(τ) = o(Γ_0(τ)).

  



  We will assume that Γ_0(τ) + Γ_-(τ) = o  (Γ_+(τ) ) for all τ≤τ_0 and obtain contradiction.
  Some of the estimates here are similar to the ones we will use in the proof of Theorem <ref> that follows next.
  We refer the reader to that proof for these details.
  Under the assumption of reflection symmetry, λ_+=1 is the only positive eigenvalue and hence, similarly to obtaining (<ref>) we can also get that
  
    Γ_+(τ) ≥ e^-1Γ_+(τ+1) - C δ(τ+1)^θ/2 Γ_+(τ+1).

  If we iterate (<ref>) and (<ref>), like in <cit.> we obtain
  
    O(e^(1+ϵ)τ) ≤Γ_+(τ) ≤ O(e^(1-ϵ)τ),

  for every ϵ > 0.  This together with (<ref>) imply
  δ(τ) ≤ e^τ/8.  Using this and iterating (<ref>) and (<ref>) again yield
  
    Γ_+(τ) = O(e^τ)         Γ(τ) = O(e^τ),

  by our assumption that Γ_0(τ) + Γ_-(τ) ≤ o(Γ_+(τ)).
  
  Recall that the eigenfunction corresponding to positive eigenvalue is h_0 = 1.  Multiply (<ref>) by h_0 and integrate over ℝ to get
  
    β_τ = β + ∫(E + E_χ + E_nl + 1/100 v χ' σ δ(τ)^-99/100δ'(τ))  dμ.

  Under our assumption about the prevailing mode, following same arguments as in the proof of Proposition <ref>
  below, we get
  
    |∫(E + E_χ + E_nl)  dμ| = O(β_*^2(τ)),

  where β_*(τ) = sup_τ' ≤τ|β(τ')|.  Similar arguments that we will use in the proof of Theorem <ref> to derive differential inequalities for α(τ), yield also that
  
    β_τ = β + O(β_*^2(τ)).

  Following the same arguments as in the proof of Theorem <ref> we can show there exists a τ_0 ≪ -1 so that for τ≤τ_0,
  
    β_τ = β + O(β^2(τ)).

  By (<ref>) we have |β(τ)| = O(e^τ) and hence,
  
    β_τ = β (1 + O(e^τ)).

  This implies
  
    β(τ) = (K + O(e^τ))  e^τ.

  We can argue as in the proof of Lemma 5.8 in <cit.> to claim that K can not vanish.  The same reparametrization arguments that we used in the proof of Lemma 5.11 in <cit.> yield a contradiction.
  This concludes the proof of Proposition <ref>.




 §.§ The proof of Theorem <ref>
 Using Proposition <ref> we are able to claim first the following result on derivatives of v̅.


  
  There exists a τ_0 ≪ -1 so that for all τ≤τ_0 we have
  
    (1-)^rv̅(τ)
      = O(α_*(τ))     (τ→-∞).

  for 0≤ r<3/2.  In particular, for r=1/2 and r=1 this implies
  
    v̅_±,σ(τ) + v̅_±,σσ(τ)
      = o(α_*(τ))
           (τ→-∞),

  where v̅_±(σ,τ) := v̅_+(σ,τ) + v̅_-(σ,τ).


At this point we can estimate the first derivative of v̅(τ).  After that, we can bootstrap and estimate (1-)^r_σv̅(τ).

  There is a constant C such that for all τ≪0 one has
  
    sup_τ'≤τv̅_σ(τ')≤ C α_*(τ).



  
  We consider the two terms in (<ref>).  The first term satisfies
  
    _σ e^v̅(τ-1)≤ Cv̅(τ-1)≤ Cα_*(τ).

  Thus (<ref>) implies
  
    v̅_σ(τ)   ≤ Cα_*(τ) + C ϵ∫_τ-1^τv̅_σ(τ') + v̅(τ')/√(τ-τ') dτ'
        + C∫_τ-1^τe^-1/64δ(τ')^-2θ/√(τ-τ') dτ' 
       ≤ Cα_*(τ) +Ce^-1/64δ(τ)^-2θ
        + Cϵsup_τ'≤τv̅_σ(τ')

  If we choose ϵ>0 so small that Cϵ≤1/2 then, after taking the supremum over τ and using the fact that δ(τ) is nondecreasing, we get
  
    sup_τ'≤τ_σv̅(τ')≤ Cα_*(τ) + C e^-1/64δ(τ)^-2θ.

  Finally, we use that for all sufficiently large τ Lemma <ref> implies that
  
    e^-1/64δ(τ)^-2θ≤ C_mδ(τ)^msup_τ'≤τ_σv̅(τ')

  so that for τ≪0 we get
  
    sup_τ '≤τ_σv̅(τ')≤ C α_*(τ),

  as claimed.



  There is a constant C such that for all τ≪0 one has for any r<1
  
    sup_τ'≤τ(1-)^rv̅_σ(τ')≤ Cα_*(τ).

  In particular, setting r=1/2 leads to
  
    sup_τ'≤τv̅_σσ(τ')≤ C α_*(τ).

  


  We consider w=v̅_σ and obtain an equation for w by differentiating (<ref>) with respect to σ.  We get
  
    w_τ -  w = (a_σ-12+b) w + aw_σ + b_σv̅ + c_σ .

  Once again we can apply the variation of constants formula to get
  
    w(τ)= e^w(τ-1) + ∫_τ-1^τ e^(τ-τ'){(a_σ-12+b) w + aw_σ + b_σv̅ + c_σ} dτ'.

  We now estimate the norm of (1-)^rw.  By choosing r>1/2 we ensure that this norm will be better than w_σ, while choosing r<1 leads to convergent integrals in the following application of the variation of constants formula.  Thus we get
  
    (1-)^rw(τ)≤
        Cw(τ-1) +
    
        C∫_τ'-1^τw(τ')+ ϵw_σ(τ') + v̅(τ') + e^-1/64δ(τ)^-2θ/(τ-τ')^r dτ' .

  Estimate w_σ(τ')≤ C (1-)^rw(τ'), and then proceed as before to conclude (1-)^rw(τ)≤ C α_*(τ).
  


We are ready now to provide the proof of Lemma <ref>.


  At this point we have shown that derivatives up to order almost 3 of v̅(τ) are bounded in terms of α_*(τ).
  To prove Lemma <ref> we have to show that v̅_±(τ)=v̅_+(τ)+v̅_-(τ) and its first and second order derivatives are o(α_*(τ)).
  This follows from an interpolation argument.
  Namely, by Proposition <ref> we know that v̅_±(τ) = o(α_*(τ)), and we have shown that (1-)^rv̅ = O(α_*(τ)) for any r<3/2.
  The interpolation inequality
  
    (1-)^s ϕ≤ϕ^1-s/r(1-)^rϕ^s/r     (0≤ s≤ r)

  implies that (1-)^rv̅(τ) = o(α_*(τ)) for r∈{1/2, 1}.
  In view of (<ref>), this implies that _σ^mv̅_±(τ) = o(α_*(τ)) for m ∈{1,2}, and therefore also completes the proof of Lemma <ref>.




 §.§ Asymptotics of α(τ)

It is easy to see that v(σ,τ) satisfies

    v_τ =
    v_σσ - σ/2  v_σ + v - v_σ^2 - v^2/2 + v v_σ^2/1+v + v^3/2(1+v) - 2 v_σ ∫_0^σv_σ^2dσ/(1+v)^2.

We also have

    v̅_τ = v̅_σσ - σ/2v̅_σ + v̅ - v̅_σ^2 -v̅^2/2 + E + E_χ + E_nl

holding for almost all τ (where δ'(τ) exits), and

    E := v̅v̅_σ^2/1+v + v̅^3/2(1+v),

is the error term containing at least quadratic and higher order terms in v̅ and its derivatives,

    E_χ   := -2v_σχ_σ
        - vχ_σσ
        + σ/2 v χ_σ
        - (1- χ)v_σv̅_σ
        + χ_σ v v̅_σ
        + χ_σ v_σ^2 
       - vv̅(1- χ)/2
        + v̅(1- χ)^2v_σ^2/1+v
        + v̅χ_σ^2v^2/1+v
        + v̅ v^2 (1- χ)^2/2(1+v)
        +v̅^2 v(1- χ)/1+v
       + 2v̅/1+v (v̅_σ v_σ (1-χ)
        - v v̅_σχ_σ
        - χ_σ (1- χ) v v_σ) + v  χ_τ

is the error term coming from introducing the cut off function χ and

    E_nl := -2 χ  v_σ∫_0^σv_σ^2dσ/(1+v)^2,

is the non-local error term.

In order to analyze the behavior of α(τ) we will multiply equation (<ref>) by h_2 and integrate it over .  We get

    α_τ =
    - ∫(v̅^2/2 + v̅_σ^2) h_2  dμ
    + ∫(E + E_χ + E_nl) h_2  dμ.

We claim the first integral in the ODE for α(τ) above is the leading order term, while the others are of lower order.  We verify this in the following proposition.



  There exists a τ_0 ≪ -1 so that for all τ≤τ_0 we have
  
    ∫(v̅^2/2 + v̅_σ^2) h_2  dμ
      = 8α(τ)^2  h_2^2
      + o(α_*(τ)^2),

  and
  
    |∫ E  h_2  dμ|
      + |∫ E_χ h_2  dμ|
      + |∫ E_nl h_2  dμ|
      = o(α_*(τ)^2),

  where the right hand sides in both integrals above are evaluated at time τ.



  Let v̅_± := _+ + _- =v̅ - α h_2.  We can write
  
    ∫(v̅^2/2 + v̅_σ^2) h_2  dμ
          =   α(τ)^2 ∫ ( h_2^3/2
          + h_2σ^2 h_2  )   dμ
       + 2α(τ) ∫ ( h_2^2 _± + (_±)_σ h_2σ h_2  )   dμ
       +∫ ( _±^2/2 +
          ( _±)_σ^2  )   h_2  dμ

  Using h_2^2 = h_4 + 8h_2 + 8h_0 and h_2σ^2 = 4h_2 + 8h_0 we compute
  
    ∫(h_2^3/2 + h_2σ^2h_2)  dμ = 8 h_2^2.

  To estimate the other terms, we recall Lemma 4.12 in <cit.> which implies that there exist universal constants C_1, C_2 so that if a function f is compactly supported in , then
  
    ∫ f^2σ^2  dμ≤ C_1 ∫ f^2  dμ + C_2 ∫ f_σ^2  dμ.

  Using this yields
  
    |∫_±^2 h_2   dμ|
      ≤ C_1 ∫_±^2  dμ
      + C_2 ∫(_±)_σ^2  dμ.

  Combining the last estimate with Lemma <ref> yields
  
    |∫_±^2h_2  dμ|
      ≤ C  sup_τ'≤τ ( _±(τ)^2
      + (_±)_σ(τ) ^2  )
      = o(α_*(τ)^2).

  Similarly,
  
    |∫(v̅_± )_σ^2 h_2  dμ| = o(α_*(τ)^2).

  By Cauchy-Schwartz
  
    ∫v̅_± h_2^2  dμ = o(α_*(τ))     ∫ (_±)_σ h_2σ h_2  dμ = o(α_*(τ)).

  By putting everything above together we obtain
  
    ∫(v̅^2/2 + v̅_σ^2) h_2  dμ
      = 8α(τ)^2 h_2^2 + o(α_*(τ)^2),

  as stated in the first part of the Proposition.
  
  To show the second estimate in the statement of the Proposition let us start with E given in (<ref>).  Recall
  that by Lemma <ref>, we have
  
    |v_σ(σ,τ)| + |v(σ,τ)| ≤ C δ(τ)^1/8,          |σ| ≤δ(τ)^-θ.

  This estimate and Lemma <ref> yield the bound
  
    | ∫ E   h_2 dμ | ≤ C sup | |  ∫ ( v̅_σ^2 + v̅^2/2 ) (σ^2 +2)  dμ
        = o(α_*(τ)^2).

  Any of the terms in (<ref>) is supported in |σ| ≥δ(τ)^-θ and hence integral of any of them is exponentially small.  That is why we treat all terms in (<ref>) the same way, so we will discuss in details only one of them.  Using that |v(σ,τ)| ≤ C δ(τ)^1/8 (Lemma 3.7 in <cit.>) and the key observation in <cit.> that
  
    δ(τ)^4 ≤ C sup_τ'≤τv(τ')^2 ≤ Cα_*(τ)^2

  (that follows by standard interpolation inequalities) we have |v(σ,τ) δ(τ)^θ h_2| ≤ C and for τ≤τ_0 ≪ -1 sufficiently small, we obtain
  
    | ∫ v χ_τ h_2  dμ | 
        =  |∫σ v χ̂' δ(τ)^θ h_2  dμ |
        ≤ C e^-1/4 δ(τ)^-2θ≤ C δ(τ)^10 
        = o(α_*(τ)^2).

  All other estimates are similar, hence concluding
  
    |∫ E_χ h_2  dμ | = o(α_*(τ)^2).

  Finally, to treat the term E_nl, we express v_σ = v̅_σ + (1- χ) v_σ - v χ_σ.  Then, using
  that χ_σ= δ(τ)^θ χ̂' and also that v_σ≤ 0 for σ≥ 0 and v_σσ≤ 0, we obtain
  
    |∫ E_nl h_2  dμ |    ≤ C ∫ |v_σ|^3 (σ^3 + 1)  dμ
       ≤ C ∫ (|v̅_σ| + (1- χ) |v_σ| + |v χ̂' |δ(τ)^θ )^3 (σ^3 + 1)  dμ.

  Once we expand the last integral on the right hand side, note that all integrals of terms that contain either χ̂' or 1-χ̂ (since those functions are supported on a set |σ| ≥δ(τ)^-θ) can be estimated by o(α_*(τ)^2), by the same reasoning as in (<ref>).  On the other hand, using (<ref>), we have
  
    ∫ |v̅_σ|^3 (σ^3 + 1)  dμ≤ Cδ^-3θ + 1/8sup_s≤τ∫v̅_σ^2   dμ 
      = o(α_*(τ)^2),

  since from our asymptotics we know sup_τ'≤τv̅_σ(τ') = O(α_*(τ)).  Hence,
  
    |∫ E_nl h_2  dμ | = o(α_*(τ)^2).

  Finally, estimates (<ref>), (<ref>) and (<ref>) conclude the proof of the Proposition.




 §.§ The conclusion of the proof of Theorem <ref>
 Using the estimates we just obtained,
we will now conclude the proof of our parabolic region asymptotics.



 
  Multiply (<ref>) by h_2 and integrate it over  with respect to measure dμ.  By Proposition <ref> we get
  
    h_2^2  α'(τ) = -8  α(τ)^2  h_2^2 + o(α_*(τ)^2) + θ δ'(τ) δ(τ)^θ-1∫ v χ' σ  dμ.

  Since δ'(τ)≥ 0 is bounded, this implies
  
    α'(τ) 
        ≤ -8 α(τ)^2 + o(α_*(τ)^2) 
        + C δ(τ)^θ-1+1/8 e^-1/4δ(τ)^-2θ

  Using Lemma <ref>, Proposition <ref> and the key observation (<ref>) from <cit.>,
  namely that δ(τ)^4 ≤ C  Γ(τ), we find that the last term is also o(α_*(τ)^2). Thus, we conclude that
  
    α'(τ) ≤ -8 α(τ)^2 + o(α_*(τ)^2).

  
  We next claim that for τ≤τ_0 ≪ -1, we have
  
    |α(τ)| = α_*(τ),        o(α_*(τ)^2) = o(α(τ)^2).

  In order to see that, it is sufficient to show that for τ_0≪ -1 sufficiently negative, |α(τ)| is monotone increasing for τ≤τ_0.
  We argue by contradiction.  If this were not true, since lim_τ→ -∞α(τ) = 0, we would be able to find a decreasing sequence of times τ_j → -∞ so that each τ_j is a local maximum for |α(τ)| and |α(τ_j)| = sup_τ'≤τ_j |α(τ')|
  =α_*(τ_j) >0.  Since α(τ_j) ≠ 0 and τ_j is local maximum for |α(τ_j)|, we
  must have α'(τ_j)=0, and by (<ref>)
  
    8 α(τ_j)^2
      = o(α_*(τ_j )^2)
      = o (α(τ_j )^2)
          
      (j→∞),

  which contradicts |α(τ_j) | >0.  We therefore conclude that (<ref>) holds.  
  
  Using what we have just shown, we conclude from (<ref>) that for τ≤τ_0 ≪ -1 we have
  
    α'(τ) ≤ - 8  α(τ)^2 + o  (α(τ)^2  ).

  Integration of this differential inequality yields
  
    1/α(τ)≤ -8   |τ|   (1+o(1)  ).

  Similarly as above, (<ref>) also yields
  
    α'(τ) ≥ -8 α(τ)^2 - o(α_*(τ)^2)

  which after integration yields
  
    1/α(τ)≥ -8   |τ|   (1+o(1)  ).

  Finally, (<ref>) and (<ref>) imply
  
    α(τ) = -1/8|τ|  (1 + o(1)).

  This concludes the proof of Theorem <ref>.




§ INTERMEDIATE REGION ASYMPTOTICS




We will assume in this section that our solution u(σ,τ) of (<ref>) satisfies the parabolic region asymptotics (<ref>).
Our goal is to derive the asymptotic behavior of u(σ,τ), as τ→ -∞, in the intermediate region

    _M,θ := {  (σ, τ) :   |σ| ≥ M      u(σ,τ) ≥ θ }.

To this end, we consider the change of variables z:=σ/√(|τ|) defining the function

    u̅(z,τ) = u(σ, τ),      z:=σ/√(|τ|).

A direct computation starting from (<ref>) shows that u̅(z,τ) satisfies the equation

    u̅_τ = 1/|τ|u̅_zz - z/2 u̅_z
      - 1/|τ|J̅(z,τ)  u̅_z + u̅_z^2/|τ| u - 1/u̅ + u̅/2.

where

    J̅(z,τ):= 2  ∫_0^z u̅_zz/u  dz.

Our goal is to prove the following result.

 
  For any θ∈ (0, √(2)), we have
  
    lim_τ→ -∞u̅(z,τ) = √(2 - z^2/2)

  uniformly for u ≥θ.  
  It follows that (<ref>) holds uniformly on |z| ≤ 2-η,
  for any η >0 small.  In particular, we have
  
    σ(τ) = 2√(|τ|) (1 +o(1)),         τ→ -∞.





Recall the change of variable Y(u,τ):=u_σ^2(σ,τ), u=u(σ,τ) which was introduced in Section <ref>.  We begin by observing that our asymptotics (<ref>) imply the following asymptotics for Y(u,τ).


   
  For any M ≫ 1 we have
  
    Y(u(σ, τ), τ) = 1/2|τ| (u(σ,τ)^-2 - 1  ) + 1/4|τ|^2 + o_M  (1/|τ|^2 )

  on |σ| ≤ M.


 We use the asymptotics (<ref>) which imply that on |σ| ≤ M,
  
    u_σ^2 = σ^2/8  |τ|^2 ( 1 + o_M (1 )  ).

  On the other hand, using also (<ref>) we have
  
    u^2 = 2  ( 1 - σ^2-2/4|τ| + o_M( 1/|τ|)  )

  implying that
  
    2/u^2 - 1 = σ^2-2/4|τ| + o_M( 1/|τ|).

  Thus, at u=u(σ, τ), we have
  
    u_σ^2 = σ^2/8  |τ|^2 ( 1 + o_M (1 )  ) =
      1/2|τ| ( 2/u^2 - 1  ) + 1/4|τ|^2 + o_M  (1/|τ|^2 ).

  




The next result follows by using the barriers constructed by S.  Brendle in <cit.> (see Proposition <ref>) and the exact behavior of Y(u,τ) at u(M,τ)
shown in (<ref>).

Assume that our solution u(σ,τ) satisfies the asymptotics (<ref>) in the parabolic region |σ| ≤ M, for all M ≫ 1, implying by Lemma <ref>, that Y(u,τ):=u_σ^2(σ,τ) with u=u(σ,τ) satisfies
  
    Y(u,τ) = 1/2|τ| ( 2 u^-2 - 1  ) + o  (1/|τ| )        u = u(M, τ) < √(2)

  and all τ≤τ_0(M) ≪ -1.
  Then, for any θ∈ (0,1) and δ >0, there exists M_δ>0 depending on δ and τ_0 = τ_0(δ, θ) ≪ -1 such that for all τ≤τ_0 ≪ -1, we have
  
    u_σ^2(σ,τ) = Y(u,τ) ≤1+δ/2|τ| ( 2u^-2 - 1  ),        θ≤ u ≤ u(M_δ, τ).





 Fix any small numbers δ >0 and θ >0 and for
  τ≤τ_0 = τ_0(δ, θ) to be determined later, consider the barrier Y_a (u)
  given in Proposition <ref> with a=a(τ) satisfying
  a^-2= 1+δ/2|τ|. For M =M(δ) ≫ 1 to be determined later set λ(τ) = u(M,τ).  Our result will follow by
  comparing Y(u,τ) with Y_a(u) using the maximum principle on the set
  
    Q_τ := ⋃_τ̅≤τ' ≤τ (  [ a^-1 r^*, λ(τ)  ] ×{τ' } ),

  for τ̅≪τ≤τ_0.  Here r^* is the universal constant as in Proposition <ref>.
  
  Lets start by showing that there exists τ_0 = τ_0(δ,θ) ≪ -1 such that for all τ≤τ_0, we have
  
    Y(λ(τ'), τ') ≤ Y_a ( λ(τ') ),          τ' ≤τ.

  To this end we will combine (<ref>) with (<ref>) from which we get the lower
  bound
  
    Y_a(u) ≥ a^-2 ( 2 u^-2 -1 ) + 1/100  a^-4

  (see Proposition 2.9 in <cit.> for this bound).
  Using this inequality we find that for u=u(M,τ')=λ(τ') and a^2=2 |τ|/(1+δ), we have
  
    Y_a( λ(τ')) ≥1+δ/2 |τ|  ( 2/λ(τ')^2 -1 )

  and on the other hand by (<ref>),
  
    Y(λ(τ'),τ') = Y(u(M, τ'), τ') = 1/2|τ'| ( 2/λ^2(τ') - 1  ) + 1/4|τ'|^2 + o_M(1/|τ'|).

  Hence, in order to guarantee that Y(λ(τ'), τ') ≤ Y_a ( λ(τ') ), it is sufficient have
  
    1/2|τ'| ( 2/λ^2(τ') - 1  ) + 1/2|τ'|^2≤1+δ/2 |τ|  ( 2/λ^2(τ') -1 )

  for all τ' ≤τ.
  But since λ(τ')=u(M,τ'), (<ref>) gives
  
    2/λ^2(τ') - 1 = M^2-2/4|τ'| + o_M( 1/|τ'|).

  Using the above and that |τ'| > |τ|, we conclude that it is sufficient to have
  
    M^2-2/ 4 |τ'|^2 + 1/2|τ'|^2 + o_M(1/|τ'|^2) ≤M^2-2/4 |τ'|^2  (1+δ) + o_M ( 1/ |τ'|^2 ),
          ∀τ'≤τ≤τ_0

  which is guaranteed, after we absorb lower order terms, if
  
    M^2-2/ 4 |τ'|^2 + 1/|τ'|^2≤M^2-2/4 |τ'|^2 (1+δ),

  or equivalently,
  M^2+2 ≤ (M^2-2)   (1+δ).  Hence, it is sufficient to choose M = M(δ) such that
  M^2+2 = (M^2-2) (1+δ), implying that
  
    M = M_δ := √(2 + 4/δ).

  For this choice of M the above inequality is satisfied for all τ' ≤τ≤τ_0(δ,θ)≪ -1.
  In all the above calculations we used the estimate o_M(1) ≤1/100, for τ' ≤τ.  This is possible with our choice of
  M_δ, as we can take τ_0=τ_0(δ) to be very negative.
  
  We will now show that for the same choice of a = a(τ), we have
  
    Y(a^-1 r^*, τ') ≤ Y_a(a^-1 r^*),     ∀ τ' ≤τ≤τ_0.

  Since our solution Y satisfies Y(u,τ) ≤ 1 always, it is sufficient to show that Y_a(a^-1 r^*) ≥ 1.  This readily follows from the construction of Y_a in <cit.>, where
  Y_a(a^-1 r^*, τ') = 2 + β_a(r^*) and 1+β_a(r^*) ≥ 1 (see in <cit.>, Proposition 2.4 for the definition of the function β_a,
  and in the proof of Proposition 2.8 for the property 1+β_a(r^*) ≥ 1).
  
  Finally, lets show that for our choice of a=a(τ) we can chose τ̅≪τ≤τ_0 ≪ -1, such that
  
    Y(u, τ̅) ≤ Y_a(u),        u ∈ [a^-1 r^*, λ(τ̅)].

  To this end, we use Lemma 2.7 in <cit.> which implies in our rescaled variables that
  
    lim inf_τ̅→ -∞sup_u ≥ a^-1 r^* Y(u, τ̅) =0.

  On the other hand, the construction of the barrier Y_a(u) in <cit.> implies that
  
    inf_u ≥ a^-1 r^* Y_a(u) =c(τ) >0

  which implies that (<ref>) holds, for some τ̅≪τ.
  
  We can now apply the comparison principle between our solution Y(u,τ) of (<ref>) and the supersolution Y_a(u)
  of the same equation on the domain Q(τ) defined
  by (<ref>), to conclude that Y(u,τ') ≤ Y_a(u) on Q(τ).  This in particular holds at τ'=τ, namely we have
  
    Y(u,τ) ≤ Y_a(τ)(u),       ∀   u ∈ [a(τ)^-1 r^*, λ(τ)].

  
  To finish the proof of the Proposition, we will bound from above Y_a(τ)(u) for u ∈ [θ, √(2)].  Recall the definition of Y_a(u) for
  u ∈ [N  a^-1, √(2)] as
  
    Y_a(u) = φ ( a u/√(2) ) - a^-2 + a^-4ζ(u)

  where φ(s) = s^-2 + O(s^-4), as s ≫ +∞ and ζ(s) bounded for s ∈ [θ, √(2)] with a bound depending on θ.
  Recalling the definition of a=a(τ) to satisfy a^2 = 2 |τ|/1+δ, we have a ≫ 1 for τ≤τ_0 ≪ -1.  Hence,
  
    Y_a(u) ≤ a^-2 (2u^-2 -1 ) + C_θ  a^-4≤1+δ/|τ| (2u^-2 -1 ) + C_θ/|τ|^2

  for all u ∈ [θ, √(2)]. Since r^*  a^-1 < θ and λ(τ)=u(M,τ) < √(2) for τ≤τ_0 ≪ -1, we finally conclude by combining (<ref>) and (<ref>) that our desired bound (<ref>) (with δ replaced by 2δ) holds on u ∈ [θ, λ(τ)] finishing the proof of the proposition.
  




We will now give the proof of our main result in this section, Proposition <ref>.

 We will show that the two bounds
  
    lim inf_τ→ +∞u̅(z,τ) ≥√(2 - z^2/2)        lim sup_τ→ +∞u̅(z,τ) ≤√(2 - z^2/2).

  hold uniformly on u ≥θ, for any θ∈ (0, √(2)).
  
  The bound from below in (<ref>):
  The desired bound
  clearly holds in the parabolic region |σ| ≤ M, for any M >0 (see Lemma <ref>).
  Hence, it is sufficient to show that for the given θ∈ (0, √(2)) and for
  any δ >0 small, there exists τ_0=τ_0(θ, δ)
  such that
  
    u(z,τ) ≥√(2 - (1+δ)  z^2/2)

  holds on the _M,θ∩{ z :    (1+δ)  z^2 <4 }, for M ≫ 1 (we need to have M ≥ M_δ,
  where M_δ is defined in Proposition <ref>).
  This will follow by integrating on [M, σ], for |σ | ≥ M ≫ 1 the bound
  
    u_σ^2(σ,τ) ≤1+δ/2|τ| ( 2  u(σ,τ)^-2 - 1  ),     τ≤τ_0(θ, δ) ≪ -1

  shown in Proposition <ref>.  Assume that σ≥ M as the case σ < -M is similar.
  Multiplying both sides of the previous inequality by u^2 and taking square roots, we obtain
  (since u_σ <0 for σ≥ M ≫ 1), the differential inequality
  
    - u u_σ/√( 2- u^2)≤√(1+δ)/√(2|τ|).

  Hence, after integration on [M, σ] we get
  
    √( 2- u^2(σ,τ)) - √( 2- u^2(M,τ))≤√(1+δ)/√(2|τ|) (σ - M).

  Next we use the parabolic region asymptotics (<ref>) which give for M ≫ 1
  
    √( 2- u^2(M,τ)) = √(M^2-2)/√(2|τ|) + o_M  ( 1/√(|τ|) ) ≤M/√(2|τ|) √(1+δ)

  and combine it with the previous estimate to obtain the bound
  
    √( 2- u^2(σ,τ) )≤√(1+δ)/√(2|τ|) (σ - M) + M/√(2|τ|) = √(1+δ) σ/√(2|τ|).

  Recalling our notation u̅(z,τ) = u(σ,τ), with z:=σ/√(|τ|), the above estimate can be written after being squared as
  
    2- u̅^2(z,τ) ≤ (1+δ)  z^2/2u̅(z,τ) ≥√( 2 - (1+δ)  z^2/2).

  The same estimate holds for z < 0
  and the bound (<ref>) is shown.  Note that the above inequality holds only when (1+δ)  z^2 <4
  and for u̅≥θ.  It follows that
  
    lim inf_τ→ -∞u̅(z,τ) ≥√( 2 - (1+δ)  z^2/2)

  and since this holds for any δ >0, we conclude the bound from below in (<ref>).
  
  The bound from above in (<ref>): This estimate follows from our equation (<ref>)
  after we use the concavity of u̅ and estimate (<ref>) to deduce from (<ref>) the first order partial differential inequality
  
    u̅_τ≤ - z/2 u̅_z (1 - κ/|τ| ) - 1/u̅ + u̅/2

  which holds on u̅≥θ, for any θ∈ (0,√(2)) and for some κ = κ(θ) >0.  Let us first verify that (<ref>) follows from our equation (<ref>) and u̅_zz≤ 0.  First, u̅_zz <0 and u_z <0 iff z >0, implies that
  
    - 1/|τ|J̅(z,τ)  u̅_z = - 2 u̅_z/|τ|∫_0^z u̅_zz  dx ≤ 0.

  Next we observe that
  
    u̅_z^2/|τ| u≤z/2 κ/|τ|        u_σ^2/u≤z/2 κ/|τ|

  for some κ = κ(θ) >0.
  Indeed, the latter estimate follows immediately by (<ref>) and (<ref>).  Using once more u̅_zz < 0, to estimate the diffusion term in (<ref>), we conclude that (<ref>)
  holds.
  
  Now let's integrate (<ref>).
  Defining :=  ^2 - 2, (<ref>) becomes equivalent to
  
    _τ≤ -z/2  (1 - κ/|τ| ) _z + .

  We see (z,τ) is a subsolution to the first order partial differential equation
  
    /τ w = -z/2  (1 - κ/|τ| )   w_z + w,

  which we can write as
  
    d/dτ w(z(τ),τ) = w(z(τ),τ),

  where
  
    d/dτ z = z/2  (1 - κ/|τ| )

  is the characteristic equation for (<ref>)
  (see Figure <ref>).
  
  Assume the curve (z(τ),τ) connects (z,τ) and (z_1,τ_1) with z_1 = M/√(|τ_1|), for M > 0 big.
  Integrate (<ref>) from τ_1 to τ to get
  
    τ_1 = τ + log(M^2/z^2|τ_1|) + κ log (|τ|/|τ_1| )

  from which it also follows that for u̅≥θ, we have
  
    τ_1 = τ  ( 1 + ϵ_1(τ)  )

  where ϵ_1(τ) <0 satisfies lim_τ→ -∞ϵ_1(τ)=0.
  At the point z_1 = M/√(|τ_1|) we can use (<ref>) to compute :
  
    (z_1,τ_1) = (z_1, τ_1)^2 - 2 = -M^2-2/2|τ_1|(1+ϵ(τ_1))

  where here and below ϵ(τ) will denote functions which may change from line to line but all satisfy lim_τ→ -∞ϵ(τ)=0.
  
  On the other hand, if we integrate (<ref>) from τ to τ_1 we get
  
    w(z,τ) = e^τ - τ_1  w(z_1,τ_1),

  and we can start w with the initial condition w(z_1,τ_1) = (z_1, τ_1), so that
  
    w(z,τ) = - z^2 |τ|/|τ_1| M^2-2/2M^2  (1 + ϵ(τ_1)),

  where we have seen hat τ_1 = τ ( 1 + ϵ_1(τ)  ).
  Therefore,
  
    w(z,τ)
      = -z^2|τ|/|τ|   (1 + ϵ(τ)  ) M^2-2/2M^2 
      (1 + ϵ_1(τ)) = -z^2  M^2-2/2M^2  (1+ ϵ̅(τ) ),

  where lim_τ→ + ∞ϵ̅(τ)=0.
  Since (z_1,τ_1) = w(z_1,τ_1), by the maximum principle applied to (<ref>), along characteristics (z(τ),τ) connecting (z_1,τ_1) and (z,τ) we have
  
    (z,τ) ≤ w(z,τ).

  This implies that for all z≥ M/√(|τ|) and as long as u̅(z,τ) ≥θ one has
  
    (z,τ) ≤√(2 - M^2-2/2M^2  z^2  (1+ ϵ(τ))),

  where again lim_τ→ -∞ϵ(τ) = 0.
  Hence, for all z∈ (0, √(2)) with u̅(z,τ) ≥θ, we have
  
    lim sup_τ→-∞(z, τ) ≤√(2 - M^2-2/2M^2  z^2).

  Since this holds for all M>0, we finally conclude the upper bound in (<ref>)
  which holds uniformly on u̅≥θ, for any θ∈ (0, √(2)).
  




§ TIP REGION ASYMPTOTICS



Recall that in commuting variables σ and τ the rescaled radius u(σ,τ) satisfies equation (<ref>).  We will use this
equation to obtain an estimate for the rescaled diameter (τ).  In our reflection symmetric case the location of the tips
is σ = σ_±(τ) = ±σ(τ), hence (τ) = σ_+(τ) - σ_-(τ) = 2σ(τ).
Since u(σ(τ),τ) = 0, after differentiating it in τ and using that u_σ(σ(τ),τ) = -1 and u_σσ(σ(τ),τ) = 0, we obtain that σ̅(τ) satisfies

    σ'(τ) = σ(τ)/2 + J(σ(τ),τ),

where we recall that J(σ,τ) is the non-local term given by (<ref>).
Using results in <cit.> we prove the following Proposition.


  
  For every ϵ > 0, there exists a τ_0 = τ_0(ϵ) ≪ -1 so that for all τ≤τ_0 we have
  
    | J(σ(τ),τ)/√(κ(τ)) - C_0| < ϵ,         C_0 := ∫_0^∞Z_0'(ρ)/ρ√(Z_0(ρ))  dρ

  where Z_0(ρ) is the Bryant soliton whose maximal curvature is equal to one, and κ(τ) is the maximum of scalar curvature of our evolving metric at time τ.



  By Corollary <ref> we know the maximum of scalar curvature is attained at σ = σ(τ) and σ = -σ(τ).  Denote by p_τ the point of the maximum of scalar curvature at time τ.  Take any sequence τ_i→ -∞ and set p_i:=p_τ_i.  Dilate our metric by κ_i = κ(τ_i) := R(p_i,τ_i).  By the compactness argument of Perelman for κ-solutions (see section 11 in <cit.>) we can extract a subsequence of dilated solutions converging to a κ-solution.  We claim that this limiting κ-solution needs to be non compact.  To show this, lets argue by contradiction and assume that the limit is compact.  This would imply there existed a uniform constant C so that √(R(p_i,τ_i)) σ(τ_i) ≤ C for all i.  By Proposition <ref> we have σ(τ_i) = 2√(|τ_i|)  (1 + o(1)) and hence we would have R(p_i,τ_i) ≤C/|τ_i| for all i.  Since p_i is the point of maximal scalar curvature at time τ_i, we would in particular have the bound R(0,τ_i) ≤C/|τ_i|, for all i.  Since R(0,τ_i) ≥1/u(0,τ_i) and the lim_i→∞ u(0,τ_i) = √(2) we immediately obtain contradiction.
  
  Once we know the limiting κ-solution is non compact, by the result in <cit.> we know that the limit is isometric to either the shrinking cylinder or the Bryant soliton whose maximal curvature is one.  Since at the tips the two curvatures K_0 and K_1 are equal we conclude that the limit must be the Bryant soliton of maximal curvature one.  This is equivalent to saying that if
  
    Z(ρ,τ) = Y(u,τ),     ρ = √(κ(τ))  u,

  then the lim_τ→ -∞ Z(ρ,τ) = Z_0(ρ), where Z_0(ρ) is the unique Bryant soliton with maximal curvature one, namely Z_0(ρ) is a solution of the equation
  
    Z_0   Z_0ρρ - 1/2  (Z_0ρ)^2 + (1 - Z_0) Z_0ρ/ρ + 2  (1-Z_0)  Z_0/ρ^2=0

  satisfying Z_0(0)=1.
  
  It was shown by R.  Bryant in <cit.> that for large and small ρ the function Z_0(ρ) satisfies the asymptotics
  
    Z_0(ρ) =
        ρ^-2 + O(ρ^-4),       ρ→∞
    
          1 - ρ^2 /6 + O(ρ^4),       ρ→0

  (see also in Appendix <ref> for the derivation of the exact constants).
  
  For any τ≪ -1, define σ_L(τ) >0 to be the number at which
  u(σ_L(τ), τ) = L√(κ(τ)) where
  κ(τ)=R(σ(τ), τ) is the scalar curvature at the tip.
  Then, we split
  
    J(σ(τ),τ) = - 2 ∫_0^σ(τ)u_σσ/u dσ' = - 2 ∫_0^σ_L(τ)u_σσ/u dσ'
      -2 ∫_σ_L(τ)^σ(τ)u_σσ/u dσ'.

  By integration by parts we have
  
    0 ≤ - ∫_0^σ_L(τ)u_σσ/u  dσ' = -∫_0^σ_L(τ)u_σ^2/u^2  dσ' - u_σ/u |_u = L√(κ(τ))≤ - u_σ/u |_u = L√(κ(τ)).

  On the other hand the convergence to the soliton implies that
  
    - u_σ/u |_u = L√(κ(τ)) = √(κ(τ))/L √( Y( L/√(κ(τ)) ,τ))≈√(κ(τ))/L √(Z_0(L) )≤ C  √(κ(τ))/L^2,

  where in the last inequality we used the soliton asymptotics (<ref>).
  Therefore we conclude that
  
    - ∫_0^σ_L(τ)u_σσ/u  dσ' ≤ C  √(κ(τ))/L^2.

  For the other integral, using the change of variables Y(u,τ) = u_σ^2(σ,τ), u=u(σ,τ) and the convergence to
  the soliton Z_0 we have
  
    -2 ∫_σ_L(τ)^u_σσ/u dσ' = ∫_0^L√(κ(τ))Y_u/u√(Y)  du
      ≈√(κ(τ)) ∫_0^LZ_0'(ρ)/ρ√(Z_0(ρ))  dρ

  where we used that
  Y_u = Z_0'(ρ) ρ_u = √(κ(τ))  Z_0'(ρ) and du/u = dρ/ρ.
  Combining the above yields that for any L ≫ 1 we have
  
    | J(σ(τ),τ)/√(κ(τ)) - C_0  |
      ≤C/L^2 +  | ∫_0^LZ_0'(ρ)/ρ√(Z_0(ρ))  dρ - C_0  |

  and since C_0 := ∫_0^∞Z_0'(ρ)/ρ√(Z_0(ρ))  dρ, we obtain
  
    | J(σ(τ),τ)/√(κ(τ)) - C_0  | ≤C/L^2 + ∫_L^∞Z_0'(ρ)/ρ√(Z_0(ρ))  dρ→ 0,       L → +∞.

  This concludes the proof of the proposition.
  


The next Proposition relates the constant C_0 with the asymptotic behavior of the Bryant soliton Z_0(ρ) of maximum scalar curvature one, as ρ→ -∞.

 Assume that Z_0 is the Bryant soliton with maximum scalar curvature at the tip equal to one.  Then,
  
    C_0 := ∫_0^∞Z_0'/ρ√(Z_0)  dρ = -1.




  
  We first observe that
  
    C_0 := ∫_0^∞Z_0'/ρ√(Z_0)  dρ = ∫_0^∞2/ρ d √(Z_0)/dρ  d ρ.

  The idea is to express the integrand 2/ρ d/dρ√(Z_0(ρ)) in divergence form, using the soliton equation (<ref>).
  This will allow us to integrate it.
  
  We observe that (<ref>) can be recognized as the pressure equation of Ψ:=√(Z_0).  Indeed, a direct calculation shows that if Z_0= Ψ^2, then
  
    Z_0   Z_0ρρ - 1/2  (Z_0ρ )^2 = Ψ^2  (2 ΨΨ_ρρ + 2 Ψ_ρ^2  ) - 1/2 (2 ΨΨ_ρ )^2 = 2 Ψ^3 Ψ_ρρ

  hence the soliton equation can be expressed in terms of Ψ as
  
    2 Ψ^3 Ψ_ρρ + 2 Ψ  (1 - Ψ^2) Ψ_ρ/ρ + 2  (1-Ψ^2)  Ψ^2/ρ^2=0.

  Dividing by 2Ψ^3 gives
  
    Ψ_ρρ + (Ψ^-2 - 1) Ψ_ρ/ρ + 1/ρ^2  (Ψ^-1-Ψ)=0.

  Next notice that
  
    d/dρ ( 1/ρ  (Ψ-Ψ^-1)  ) = - 1/ρ^2  (Ψ-Ψ^-1) +(Ψ^-2 + 1) Ψ_ρ/ρ

  which gives
  
    1/ρ^2  (Ψ^-1-Ψ) = d/dρ ( 1/ρ  (Ψ-Ψ^-1)  ) - (Ψ^-2 + 1) Ψ_ρ/ρ.

  Substituting this into (<ref>) yields that the soliton equation can be expressed in terms of Ψ as
  
    Ψ_ρρ + d/dρ ( 1/ρ  (Ψ-Ψ^-1)  ) - 2 Ψ_ρ/ρ =0

  implying that
  
    2/ρ Ψ_ρ = d/dρ ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  ).

  
  Now using (<ref>) we obtain
  
    C_0    := ∫_0^∞2/ρ d √(Z_0)/dρ  d ρ = ∫_0^∞2/ρ Ψ_ρ  d ρ = ∫_0^∞d/dρ ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  )  dρ
        = lim_ρ→ +∞ ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  ) - lim_r→ 0 ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  ).

  By the Bryant soliton asymptotics (<ref>) it is easy to see that
  
    lim_r→ 0 ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  ) =0,   lim_r→ +∞ ( Ψ_ρ + 1/ρ  (Ψ-Ψ^-1)  ) = - 1

  thus concluding that
  C_0:= -1, as claimed.


As a corollary of the previous two Propositions we next compute the maximum rescaled scalar curvature.


   The maximum rescaled scalar curvature κ(τ) of our solution satisfies the asymptotic behavior

    κ(τ) = |τ| (1 + o(1)),       τ→ -∞.

  



  On the one hand by Propositions <ref> and <ref> we have
  
    J(σ(τ),τ) = - √(κ(τ))  (1 + o(1)).

  On the other hand, the intermediate region asymptotics discussed in section <ref> imply that
  
    σ(τ) = 2 √(|τ|)  (1 + o(1)),       τ→ -∞.

  Hence by (<ref>)
  
    σ'(τ)/σ(τ) = 1/2 + J(σ(τ),τ)/σ(τ) = 1/2 - √(κ(τ))  (1+o(1))/2 √(|τ|).

  Integrating this from τ to τ+ϵ, for any small ϵ > 0 yields
  
    lnσ(τ+ ϵ)/σ(τ) = ϵ/2 -  (1 + o(1))/2 √(|τ|) ∫_τ^τ+ϵ√(κ(τ'))  dτ'.

  Using also (<ref>) we obtain
  
    -ϵ/2|τ|  (1+o(1)) = ϵ/2 - (1 + o(1))/2√(|τ|) ∫_τ^τ+ϵ√(κ(τ'))  dτ'.

  Divide by ϵ and then let ϵ→ 0 above, to conclude that (<ref>) holds.


Using the above result and similar arguments as in the proof of Theorem 1.6 in <cit.>, we obtain the following convergence theorem which we express in
terms of the unrescaled solution (M, g(t)).


  Denote by p_t the point of maximal scalar curvature of (M,g(t)).  For any t < 0 define the rescaled metric
  
    g̃_t(·,s) = k(t)   g(·,t+k(t)^-1  s),

  where k(t) = R_max(t) = R(p_t,t). Then
  
    k(t) = (1 + o(1)) log|t|/|t|,        t → -∞.

  Moreover, the family of rescaled solutions (M,g̃_t(·,s), p_t) to the Ricci flow, converges to the unique Bryant soliton of maximal scalar curvature equal to one,
  namely the unique rotationally symmetric steady soliton with maximal scalar curvature equal to one.



  The proof follows by the same arguments as in the proof of Theorem 1.6 in <cit.>, using Corollary <ref> and Hamilton's Harnack estimate for the Ricci flow shown in <cit.>.






§ BRYANT SOLITON




In <cit.> Bryant showed that up to constant multiples, there is only one complete, steady, rotationally symmetric soliton in dimension three that is not flat.  It has positive sectional curvature which reaches the maximum at the center of rotation.  He showed that if we write the complete soliton (whose maximal sectional curvature is 1/6) in the form g = ds̅^2 + f(s̅)^2 g_S^2, where s̅ is the distance from the center of rotation, one has, for large s̅ the following asymptotics: the aperature a(s̅) has leading order term √(2s̅), the radial sectional curvature has leading order term 1/4s̅^2 and the orbital sectional curvature has leading order term 1/2s̅.  Note that in our old notation we have s = s(t) - s̅, where s(t) is the distance from the origin to the center of rotation (or the point of maximal scalar curvature).

As it is explained at the end of Section <ref>, around the point of maximal scalar curvature (which in this case coincides with the center of rotation, see Corollary <ref>), it is more convenient to write the metric in the form f_s^-2  df^2+ f^2 g_S^2, where Z = f_s^2 is a function of f.  In the case of a steady Ricci soliton, the profile function Z is known to satisfy the equation

    Z Z_ff - 1/2 Z_f^2 + (1 -Z) Z_f/f + 2(1-Z) Z/f^2 = 0.

Moreover, it is known that Z(f) has the following asymptotics.  Near f = 0, Z is smooth and has the asymptotic expansion

    Z(f) = 1 + b_0   f^2 + 2/5  b_0^2   f^4 + o(f^4),

where b_0 < 0 is arbitrary.  As, f → + ∞, Z is smooth and has the asymptotic expansion

    Z(f) = c_0   f^-2 + 2 c_0^2   f^-4 + o(f^-4),

where c_0 > 0 depends on b_0.



We will next find (for the convenience of the reader) the exact values of the constants b_0 and c_0 in the above asymptotics
for the Bryant soliton of maximal scalar curvature one.  These exact asymptotics were used in Section <ref>.


Recall that the scalar curvature R = 4K_0 + 2K_1.  By Corollary <ref> we know that the maximal scalar curvature is attained at s̅ = 0, at which K_0 = K_1 and hence R = 6 K_0.  An easy consequence of Lemma <ref> is that the maximal scalar curvature being equal to one is equivalent to the maximal sectional curvature being equal to 1/6.  Bryant's asymptotics
imply that for s̅ is sufficiently large, the aperture f ≈√(2s̅), implying that s̅≈f^2/2.  We also have the radial sectional curvature K_0 ≈ 1/4s̅^2 ≈ f^-4, for a ≫ 1.  On the other hand, using (<ref>) we also have

    K_0 = -Z_f/2f≈c_0/f^4,         a≫1

implying that c_0 = 1.
We also have at f = 0 (which corresponds to a point of maximal sectional curvature),
1/6 = K_1 =(1-Z)/a^2 ≈ -b_0, for a ≪1,
implying that b_0 = -1/6.



Summarizing the above discussion we conclude the following asymptotics for the Bryant soliton with maximal scalar curvature equal to
one:

    Z(f)=  1 - f^2/6 + O(f^4),              f→ 0 
    
        f^-2  + O(f^-4),             f→∞.






§ PROPERTIES OF A ROTATIONALLY SYMMETRIC SOLUTION




We assume throughout this section that ψ(·, s) is a solution to equation (<ref>) or equivalently
u(·, τ) is a solution of (<ref>).  We will prove some crucial geometric properties which we used
in the proof of our main result in this paper.  We remark that we do not need to assume reflection symmetry.  Let us recall that we have denoted by s_±(t) the tips of
our solution M_t which in rescaled variables are denoted by σ_±(τ).  We will analyze our solution for s∈ [0,s_+(t)] (or σ∈ [0,σ_+(τ)]), since the solution for s∈ [s_-(t),0] (or σ∈ [σ_-(τ),0]) can be treated similarly.

We define next the scaling invariant quantity

    Q:= K_0/K_1 = -ψψ_ss/1 - ψ_s^2 = -u u_σσ/1 - u_σ^2.

In this section we will prove Q ≤ 1 everywhere on our manifold and that the maximum scalar curvature is attained at the tips σ = σ_±(τ).
A direct calculation shows that Q satisfies the equation

    Q_t = Q_ss - 2ψ_s/ψ (1-2Q)   Q_s + 2(1-Q)/ψ^2 ( (1-ψ_s^2)   Q^2 + ψ_s^2   (2Q + 1)).


Note that the closing conditions (<ref>) imply that we can apply l'Hospital's rule to conclude

    K_1(s_+,t) = lim_s→ s_+(-ψ_ss/ψ) = K_0(s_+,t)

for all t∈ (-∞,T).  Similarly we have K_1(s_-,t) = K(s_-,t) for all t∈ (-∞,T).
Hence, Q(s_-,t) = Q(s_+,t) = 1, for all t∈ (-∞,T).


  
  Let (S^3, g(t)) be any rotationally symmetric compact, κ-noncollapsed ancient solution to the Ricci flow on S^3.
  Then, Q ≤ 1 on S^3, for all t∈ (-∞,T).



  If (S^3, g(t)) is isometric to the sphere, then our claim clearly holds.  Hence, we may assume that the rescaled backward limit of our solution is the cylinder.
  
  We have just observed that at both poles s_± we have Q = 1.  Hence, Q_max(t) is achieved on the surface
  and Q_max(t) ≥ 1.  We will now show that Q_max(t) ≤ 1.
  At a maximum of Q we have Q_s = 0 and Q_ss≤ 0 so that
  
    dQ_max/dt≤ - 2/ψ^2(Q_max - 1) ((1-ψ_s^2) Q_max^2 + ψ_s^2 (2Q_max + 1)).

  In particular this shows that if Q_max(t_0) > 1 for some t_0 < T, then the same holds for all t ≤ t_0.
  
  Since the rescaled solution u(·, τ) converges to √(2(n-1)) as τ→-∞ and it is concave,
  we have u(·, τ) ≤ C, for all τ≤τ_0.  Hence, ψ(·, t)≤ C √(T-t), for all t≤ t_0 ≪ -1,
  implying that Q_max as a function of τ, satisfies
  
    d(Q_max-1)/dτ≤ -c   (Q_max - 1) ((1-u_σ^2) Q_max^2 + u_σ^2 (2Q_max + 1)),

  where we have used that ψ_s=u_σ.
  If Q_max is attained at a point where u_σ^2 ≤ 1/2, then using that Q_max > 1
  
    d/dτ (Q_max - 1) ≤ -c   (Q_max - 1)   Q_max^2 ≤ -c   (Q_max - 1)^2,

  where constant c may change from line to line, but is uniform, independent of τ.  On the other hand, if Q_max is attained at a point where 1/2≤ u_σ^2 ≤ 1, then
  
    d/dτ (Q_max - 1) ≤ -c   (Q_max - 1) (2Q_max +1) ≤ - c  (Q_max - 1)^2.

  This means that we actually have (<ref>) holding for all τ≤τ_0, and thus
  
    d/dτ ( 1/Q_max - 1 ) ≥ c,        τ≤τ_0 ≪ -1.

  As τ→ -∞ this leads to contradiction with Q_max - 1 > 0.



  
  For each τ, the rescaled scalar curvature R(·,τ) achieves its maximum at σ = ±σ(τ).



  By Lemma <ref> we have that K_0/K_1≤ 1 on S^3, for all τ.  This implies
  
    u u_σσ - u_σ^2 + 1 ≥ 0,          (K_1)_σ≥ 0,

  for σ≥ 0, since in that region u_σ≤ 0.  Using that observation we conclude that for the scalar curvature, at any point 0 ≤σ < σ_+ and any τ, we have
  
    R(σ,τ) = 4K_0(σ,τ) + 2 K_1(σ,τ) ≤ 6K_1(σ,τ) ≤ 6K_1(σ_+,τ) = R(σ_+,τ).

  The case σ <0 can be shown similarly.


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