# Limit lamination theorems for H-surfaces

William H. Meeks III and Giuseppe Tinaglia

# Abstract

In this paper we prove some general results for constant mean curvature lamination limits of certain sequences of compact surfaces Mn embedded in 3 with constant mean curvature Hn and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non-zero constant mean curvature case, similar structure theorems by Colding and Minicozzi in [6, 8] for limits of sequences of minimal surfaces of fixed finite genus.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1309236

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/M024512/1

Funding statement: This material is based upon work for the NSF under award no. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. The second author was partially supported by EPSRC grant no. EP/M024512/1.

## A Appendix

In this appendix we give the definition of a weak CMC lamination of a Riemannian three-manifold. Specializing to the case where all of the leaves have the same mean curvature H, one obtains the definition of a weak H-lamination, for which we give a few more explanations. A simple example of a weak 1-lamination of 3 that is a not a 1-lamination is the union of two spheres of radius 1 that intersect at single point of tangency.

For further background material on these notions see [22, Section 3], [21] or our previous papers [28, 29].

## Definition A.1.

A (codimension-one) weak CMC lamination of a Riemannian three-manifold N is a collection {Lα}αI of (not necessarily injectively) immersed constant mean curvature surfaces called the leaves of , satisfying the following four properties.

1. (i)

αILα is a closed subset of N. With an abuse of notation, we will also consider to be the closed set αILα.

2. (ii)

The function |A|:[0,) given by

|A|(p)=sup{|AL|(p)L is a leaf of  with pL}

is uniformly bounded on compact sets of N.

3. (iii)

For every pN, there exists an εp>0 such that if for some αI, qLαBN(p,εp), then Lα contains a disk neighborhood of q.

4. (iv)

If pN is a point where either two leaves of intersect or a leaf of intersects itself, then each of these surfaces nearby p lies at one side of the other (this cannot happen if both of the intersecting leaves have the same signed mean curvature as graphs over their common tangent space at p, by the maximum principle).

Furthermore, if N=αLα, then we call a weak CMC foliation of N. If the leaves of have the same constant mean curvature H, then we call a weak H-lamination of N (or H-foliation, if additionally N=αLα).

### Figure 6

The leaves of a weak H-lamination with H0 can intersect each other or themselves, but only tangentially with opposite mean curvature vectors. Nevertheless, on the mean convex side of these locally intersecting leaves, there is a lamination structure.

The following proposition follows immediately from the definition of a weak H-lamination and the maximum principle for H-surfaces.

## Proposition A.2.

Any weak H-lamination L of a Riemannian three-manifold N has a local H-lamination structure on the mean convex side of each leaf. More precisely, given a leaf Lα of L and given a small disk ΔLα, there exists an ε>0 such that if (q,t) denotes the normal coordinates for expq(tηq) (here exp is the exponential map of N and η is the unit normal vector field to Lα pointing to the mean convex side of Lα), then the exponential map exp is an injective submersion in U(Δ,ε):={(q,t)qInt(Δ),t(-ε,ε)}, and the inverse image exp-1(L){qInt(Δ),t[0,ε)} is an H-lamination of U(Δ,ε) in the pulled back metric, see Figure 6.

## Definition A.3.

A leaf Lα of a weak H-lamination is a limit leaf of if at some pLα, on its mean convex side near p, it is a limit leaf of the related local H-lamination given in Proposition A.2.

## Remark A.4.

1. (i)

A weak H-lamination for H=0 is a minimal lamination.

2. (ii)

Every CMC lamination (resp. CMC foliation) of a Riemannian three-manifold is a weak CMC lamination (resp. weak CMC foliation).

3. (iii)

Theorem 4.3 in [20] states that the 2-sided cover of a limit leaf of a weak H-lamination is stable. By [25, Lemma 3.3] and the main theorem in [36], the only complete stable H-surfaces in 3 are planes. Hence, every leaf L of a weak H-lamination of 3 is properly immersed and has an embedded half-open regular neighborhood N(L) on its mean convex side, and N(L) can be chosen to be disjoint from if L is not a plane. In particular, if L is a leaf of a weak H-lamination of 3, then there is a small perturbation L of L in N(L) that is properly embedded in 3.

# Acknowledgements

The authors would like to thank Joaquin Perez for making Figure 2.

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