A note on minimal surfaces with bounded index
Davi Maximo


For any closed Riemannian three-manifold, we prove that for any sequence of closed embedded minimal surfaces with uniformly bounded index, the genus can only grow at most linearly with respect to the area.







Indication of Another Intermediate-mass Black Hole in the Galactic Center
    Mariko Nomura
    
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§ INTRODUCTION


There has been recently spectacular progress on the existence theory of minimal hypersurfaces led by the landmark work of F. Marques and A. Neves on the Almgren-Pitts minmax theory <cit.> and the volume spectrum <cit.> (jointly with Y. Liokumovich). For instance, by Marques-Neves <cit.> and A. Song <cit.>, we now know that any closed Riemannian manifold of dimension 3≤ n≤ 7 contains infinitely many smoothly embedded closed minimal hypersurfaces. In addition, when the metric on M is generic, Irie-Marques-Neves <cit.> showed that the union of embedded minimal hypersurfaces is dense. Soon after,  Marques-Neves-Song <cit.> proved that there is a sequence of such minimal hypersurfaces that is actually equidistributed. Finally, in 3 dimensions and again for generic metrics,  Chodosh-Mantoulidis <cit.> showed the existence of minimal surfaces with arbitrarily large area, genus, and Morse index.

It is interesting to ask in general how the index, topology, and geometry (area and curvature) of minimal surfaces relate to each other. The answer can be subtle. For instance, in presence of positive Ricci curvature, Choi–Schoen <cit.> proved that a sequence of minimal surfaces with bounded genus must have bounded area, curvature, and index. In a general Riemannian three-manifolds, however, it is not possible to bound the area nor the index of an embedded minimal surface by the genus, even if one assumes positive scalar curvature, as it can be seen in examples constructed by Colding–De Lellis <cit.>. On the other hand, jointly with O Chodosh and D. Ketover <cit.>, we were able to show that manifolds positive scalar curvature, uniform index bounds do imply uniform area and genus bounds. However, without ambient curvature assumptions, there can be sequences of minimal surfaces with uniformly bounded index but arbitrarily large genus and area <cit.>.
 
The purpose of this note is to prove the following estimate:

Let (M^3,g) be a fixed closed Riemannian three-manifold. For any natural number I, there exists a constant C=C(M,g,I)>0 such that if Σ_j is a sequence of two-sided, closed, connected, embedded minimal surfaces with (Σ_j)≤ I, then
  
    (Σ_j)<C (Σ_j).
 


The proof of Theorem <ref> relies on the local structure and surgery theorems proven in <cit.> which give a complete picture of how a sequence with uniformly bounded index can degenerate. Tools from there-manifold topology, namely methods of laminations and branched surfaces, are also key.


	The proof of Theorem <ref> can be modified by routine arguments, as in <cit.>, so that the result remains true for one-sided minimal surfaces.




The estimate in Theorem <ref> is essentially sharp as seen by the following example. In <cit.>, Jaco constructed a sequence of incompressible surfaces S_k in S ×^1, where S is a closed oriented surface of genus r>1. These surfaces have the property that, for each k, the projection p:S ×^1→ S restricted to S_k is a covering map of S of order k. In particular, their Euler characteristic satisfy χ(S_k)=kχ(S), so the genus of S_k grows linearly in k.

Fixing a constant curvature metric in S, we may then minimize area  in each homotopy class of Σ_k in S ×^1 using <cit.> and obtain as a result a stable minimal surface Σ_k which is embedded (after passing to a one-sided quotient, if necessary) by <cit.>.  These will have uniformly bounded curvature in k, by <cit.>, and thus area growth at least linear in k since p|_Σ_k: Σ_k → S is a k-cover.



Colding–Minicozzi <cit.> showed that there is a C^2-open set of metrics g on any manifold M so that there is a sequence of embedded stable minimal tori Σ_j with _g(Σ_j)→∞ and thus the reverse inequality in Theorem <ref> does not hold in general. 



The estimate in Theorem <ref> seems to be new in the literature even in the particular case of stable minimal surfaces.  



	H. Rosenberg and B. Meeks have told me they have a different approach to obtain a similar bound to Theorem <ref>.




 §.§ Acknowledgements
 I would like to thank H. Rosenberg for asking me whether such bound were true. I would also like to thank  O. Chodosh and J. Nogueira for interesting discussions about laminations.





§ PRELIMINARIES




 §.§ Minimal laminations
 Minimal laminations are a natural generalization of minimal surfaces. They are very convenient in dealing with a sequence of surfaces without area bounds. We recall their definition:



A closed set  of M^3 is called a minimal lamination if it is the union of pairwise disjoint, connected, injectively immersed minimal surfaces, which we call leaves. For each point x∈, we ask for the existence of a neighborhood Ω of x and a C^0,α local coordinate chart Φ:Ω→ℝ^3 under which image the leaves of  pass through in slices of the form ℝ^2×{t}∩Φ(Ω).



As an illustration of how laminations naturally appear when dealing with embedded minimal surfaces with no area bounds, consider a sequence of closed stable minimal surfaces Σ_j on (M^3,g). Suppose their areas are blowing up, i.e., |Σ_j|→∞[Such sequence of stable minimal surfaces with unbounded area appear, for example, in every metric of the 3-torus, see Example 1.13 of <cit.>]. By curvature estimates of Schoen <cit.>, Σ_j must have uniformly bounded second fundamental form, that is: 

    |II_Σ_j|≤ C

for some C>0. Thus, after a subsequence, Σ_j converges locally smoothly to a minimal lamination  (see, for instance, page 475 by <cit.>). 




 §.§ Branched surfaces
 To analyze laminations that might appear as a sequence of minimal surfaces, we will consider branched surfaces that carry them. We remark that such methods were pioneered by Williams <cit.> and have been successfully used in geometry topology on several occasions, e.g. see <cit.> and, more recently, <cit.>. They are higher dimensional analogs of train tracks carrying geodesic laminations, see <cit.>.



A branched surface B on a three-manifold M is a two-complex consisting of a finite union surfaces that combine along the 1-skeleton giving it a well-defined tangent space at every point, and generic singularities. Every point p ∈ B thus has a neighborhood in
M which is homeomorphic to one of the local models in Figure <ref>:


For branched surface B, we define its branched locus to be the set of points in B which do not have a neighborhood diffeomorphic to ℝ^2. We call the closure of each component of B∖ L a branch sector. As with train tracks, a branched surface has a well-defined normal bundle in a 3-manifold and for any ε>0 sufficiently small, an ε-tubular neighborhood N_ε(B) can be foliated by intervals transverse to B as an I-bundle in such a way that collapsing these intervals collapses N(B) to a new branched surface which can be canonically identified with B. 

We say that a surface Σ (or, similarly, a lamination ) is fully carried by B if there exists ε>0 such that S ⊂ N_ε(B) transversely intersects every I-fiber of N_ε(B). If π:N(B)→ B is the projection that collapses every I-fiber to a point and b_1,…,b_N are the components of B∖ L, we let x_i= |Σ∩π^-1(b_i)| for each b_i. We can describe Σ combinatorially via a non-negative integer coordinate (x_1,…,x_N)∈ℝ^N satisfying an obvious system of branch equations coming the from intersections of the respective branch sectors , see <cit.>:








Every lamination  in M is fully carried by a branched surface B.


This follows by general topological arguments, see <cit.>. However, we sketch an argument for the laminations relevant to the proof of Theorem <ref>.

Suppose  is a minimal lamination which is the limit of a sequence of closed embedded minimal surfaces Σ_j with |II_Σ_j|≤ C for some fixed C>0. Following a standard argument on page 475 of <cit.>, we may find r>0 sufficiently small and cover  with extrinsic balls B_r(x_1),B_r(x_2),…,B_r(x_k), where x_1,x_2…,x_k  belong to , such that the intersection of any leaf with any of the balls B_r(x_i) is either empty or the graph of a function with small gradient over a subset of a disk passing through x_i. Such discs can be the isotoped and glued to form the desired branched surface.
	

We remark that the argument above implies the constructed branched surface B also carry the surfaces Σ_j or j sufficiently large. In fact, Σ_j ⊂ N_2r(B).









 §.§ Local structure and surgery theorems
 In Remark <ref>, we saw that a sequence of stable minimal surfaces converges, locally smoothly and after passing to a subsequence, to a minimal lamination. In this section we recall the main results of <cit.>, which guarantees that closed embedded minimal surfaces with uniformly bounded index behave similarly qualitatively, up to controlled errors. What follows comprises of combining Theorem 1.17 and Corollary 1.19 of <cit.>:


There exist functions r̃(I) and m̃(I) with the following property. Fix a closed three-manifold (M^3,g) and a natural number I ∈. Suppose Σ_j⊂ (M,g) is a sequence of closed embedded minimal surfaces with (Σ_j)≤ I. Then, after passing to a subsequence, there is C>0 and a finite set of points _j⊂Σ_j with cardinality |_j|≤ I so that the curvature of Σ_j is uniformly bounded away from the set _j, i.e.,

    |_Σ_j|(x) min{1,d_g(x,_j)}≤ C,

but not at _j, i.e.,

    lim inf_j→∞min_p∈_j|_Σ_j|(p) = ∞.

Passing to a further subsequence, the points _j converge to a set of points _∞ and the surfaces Σ_j converge locally smoothly, away from _∞, to some lamination ⊂ M ∖_∞. The lamination has removable singularities, i.e., there is a smooth lamination ⊂ M so that = ∖_∞. Moreover, there exists ε_0>0 smaller than the injectivity radius of (M,g) so that _∞ is 4ε_0-separated and for any ε∈ (0,ε_0], taking j sufficiently large guarantees that there exists embedded surfaces Σ_j⊂ (M^3,g) satisfying:

	
  * The new surfaces Σ_j agree with Σ_j outside of B_ε(_∞). 
	
  * The components of Σ_j∩ B_ε(_∞) that do not intersect the spheres ∂ B_ε(_∞) transversely and the components that are topological disks appear in Σ_j without any change.
	
  * The curvature of Σ_j is uniformly bounded, i.e.
	
    lim sup_j→∞sup_x∈Σ_j|_Σ_j|(x) <∞.

	
  * Each component of Σ_j∩  B_ε(_∞) which is not a component of Σ_j∩ B_ε(_∞) is a topological disk of area at most 2πε^2(1+o(ε)).
	
  * The genus drops in controlled manner, i.e.,
	
    (Σ_j)-r̃(I) ≤(Σ_j) ≤(Σ_j).

	
  * The number of connected components increases in a controlled manner, i.e.,
	
    |π_0(Σ_j)|≤ |π_0(Σ_j)| ≤ |π_0(Σ_j)| + m̃(I).

	
  * While Σ_j is not necessarily minimal, it is asymptotically minimal in the sense that lim_j→∞‖ H_Σ_j‖_L^∞(Σ_j) = 0.

Finally, the new surfaces Σ_j converge locally smoothly to the smooth minimal lamination .
 



§ END OF PROOF OF THEOREM <REF>


We argue by contradiction: if the desired constant does not exist, we may find a sequence Σ_j of closed, two-sided, embedded minimal surfaces with (Σ_j)≤ I and such that:
    (Σ_j)/(Σ_j)↗∞.

By Theorem 1.1 of <cit.>, this implies (Σ_j)→∞; otherwise, bounded index and bounded area would imply bounded genus and thus contradict (<ref>).

Passing to a subsequence if necessary, we may then apply Theorem <ref> to Σ_j and obtain, after surgery, a sequence of nearly minimal surfaces Σ_j which, by (4), (5), (6), we may also assume are connected and  satisfy:

    (Σ_j)/(Σ_j)↗∞.

This crucially uses the fact that the surgery procedure deletes at most I components and replaces them with disks of comparable area. 


Moreover, again by Theorem <ref>, Σ_j converges locally smoothly to a smooth minimal lamination . By Proposition <ref>, we can find a branched surface B that fully carries . As in Section <ref>, let L be the branched locus of B and b_1,b_2,…,b_N its branched sections, i.e., the components of B∖ L. Each Σ_j then correspond to an non-negative integer coordinate:

    Σ_j ⟶ (x^j_1,x^j_2…,x^j_N)∈ℝ^N.

Moreover, by Remark <ref>, Σ_j ⊂ N_2r(B) for j sufficiently large, and it may be reconstructed from disks which are graphical over the sectors b_1,…,b_N. Thus, 

    x^j_1|b_1|+ x^j_2|b_2| +⋯ +x^j_N|b_N| = O(|Σ_j|),

where |b_i| is the area of the branch sector b_i. 

On other hand, the Euler characteristic of can also be estimated using a triangulation obtained by gluing the graphical pieces over the sectors b_1,…,b_N. This yields the estimate

    |χ(Σ_j)| = O(x_1+x_2+…+x_n)

which contradicts equation (<ref>) and we are done.







 
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