In this paper we prove some general results for constant mean curvature lamination limits of certain sequences of compact surfaces embedded in with constant mean curvature 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.
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 , 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 that is a not a 1-lamination is the union of two spheres of radius 1 that intersect at single point of tangency.
A (codimension-one) weak CMC lamination of a Riemannian three-manifold N is a collection of (not necessarily injectively) immersed constant mean curvature surfaces called the leaves of , satisfying the following four properties.
is a closed subset of N. With an abuse of notation, we will also consider to be the closed set .
The function given by
is uniformly bounded on compact sets of N.
For every , there exists an such that if for some , , then contains a disk neighborhood of q.
If 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 , 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 ).
The following proposition follows immediately from the definition of a weak H-lamination and the maximum principle for H-surfaces.
Any weak H-lamination 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 of and given a small disk , there exists an such that if denotes the normal coordinates for (here is the exponential map of N and η is the unit normal vector field to pointing to the mean convex side of ), then the exponential map is an injective submersion in , and the inverse image is an H-lamination of ) in the pulled back metric, see Figure 6.
A leaf of a weak H-lamination is a limit leaf of if at some , on its mean convex side near p, it is a limit leaf of the related local H-lamination given in Proposition A.2.
A weak H-lamination for is a minimal lamination.
Every CMC lamination (resp. CMC foliation) of a Riemannian three-manifold is a weak CMC lamination (resp. weak CMC foliation).
Theorem 4.3 in  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 , the only complete stable H-surfaces in are planes. Hence, every leaf L of a weak H-lamination of is properly immersed and has an embedded half-open regular neighborhood on its mean convex side, and 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 , then there is a small perturbation of L in that is properly embedded in .
The authors would like to thank Joaquin Perez for making Figure 2.
 M. Berger, A panoramic view of Riemannian geometry, Springer, Berlin 2003. Search in Google Scholar
 J. Bernstein and C. Breiner, Conformal structure of minimal surfaces with finite topology, Comm. Math. Helv. 86 (2011), no. 2, 353–381. Search in Google Scholar
 T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks, Ann. of Math. 160 (2004), 27–68. 10.4007/annals.2004.160.27Search in Google Scholar
 T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks, Ann. of Math. 160 (2004), 69–92. 10.4007/annals.2004.160.69Search in Google Scholar
 T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. III. Planar domains, Ann. of Math. 160 (2004), 523–572. 10.4007/annals.2004.160.523Search in Google Scholar
 T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. IV. Locally simply-connected, Ann. of Math. 160 (2004), 573–615. 10.4007/annals.2004.160.573Search in Google Scholar
 T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. V. Fixed genus, Ann. of Math. 181 (2015), no. 1, 1–153. 10.4007/annals.2015.181.1.1Search in Google Scholar
 P. Collin, Topologie et courbure des surfaces minimales de , Ann. of Math. (2) 1451 (1997), 1–31. Search in Google Scholar
 M. do Carmo, Riemannian geometry, Birkhäuser, Boston 1992. Search in Google Scholar
 M. do Carmo and C. K. Peng, Stable complete minimal surfaces in are planes, Bull. Amer. Math. Soc. 1 (1979), 903–906. Search in Google Scholar
 D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199–211. 10.1002/cpa.3160330206Search in Google Scholar
 N. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465–503. 10.4310/jdg/1214443598Search in Google Scholar
 R. Kusner, Global geometry of extremal surfaces in three-space, Ph.D. thesis, University of California, Berkeley, 1988. Search in Google Scholar
 W. H. Meeks III, J. Pérez and A. Ros, The geometry of minimal surfaces of finite genus. I. Curvature estimates and quasiperiodicity, J. Differential Geom. 66 (2004), 1–45. 10.4310/jdg/1090415028Search in Google Scholar
 W. H. Meeks III, J. Pérez and A. Ros, Stable constant mean curvature surfaces, Handbook of geometrical analysis, Adv. Lect. Math. 7, International Press, Somerville (2008), 301–380. Search in Google Scholar
 W. H. Meeks III, J. Pérez and A. Ros, Properly embedded minimal planar domains, Ann. of Math. 181 (2015), no. 2, 473–546. Search in Google Scholar
 W. H. Meeks III and G. Tinaglia, The dynamics theorem for surfaces in , J. Differential Geom. 85 (2010), 141–173. Search in Google Scholar
 W. H. Meeks III and G. Tinaglia, Existence of regular neighborhoods for H-surfaces, Illinois J. of Math. 55 (2011), no. 3, 835–844. Search in Google Scholar
 W. H. Meeks III and G. Tinaglia, The rigidity of embedded constant mean curvature surfaces, J. reine angew. Math. 660 (2011), 181–190. Search in Google Scholar
 W. H. Meeks III and G. Tinaglia, The geometry of constant mean curvature surfaces in , preprint (2016), https://nms.kcl.ac.uk/giuseppe.tinaglia/research.html. Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston