Limit lamination theorems for H-surfaces

Abstract

In this paper we prove a theorem concerning lamination limits of sequences of compact disks M n embedded in R 3 with constant mean curvature H n , when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi [8]. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in R 3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi in [9] for minimal disks.

The local picture theorem on the scale of topology

“…One can travel quickly up and down the horizontal levels of the limiting surfaces only along the helicoidal columns in much the same way that some parking garages are configured for traffic flow; hence, the name parking garage structure. Parking garage structures also appear as natural objects in the main results of the papers [11,13,39].…”

“…Section 5 includes some applications of Theorem 1.1. We refer the reader to [23,24,25,26,27,36,38,40] for further applications of Theorem 1.1.…”

Triply periodic constant mean curvature surfaces

“…Either M n has locally bounded norm of the second fundamental form in R 3 or not. If M n does NOT have locally bounded norm of the second fundamental form in R 3 then by Theorem 1.5 in [22], the surfaces M n converge on compact subsets of R 3 to a minimal parking garage structure of R 3 with two oppositely oriented columns, see Figure 1 and see for instance [12] for a detailed description of this limit object. Figure 1: Parking garage structure: in this picture, the sequence of surfaces on the left hand side converge smoothly away from the union S 1 ∪ S 2 or two straight lines orthogonal to the foliation of horizontal planes described on the right hand side.…”

“…If M n has locally bounded norm of the second fundamental form in R 3 then by applying Theorem 1.3 in [22] and after replacing by a subsequence, the surfaces M n converge with multiplicity one or two on compact subsets of R 3 to a properly embedded minimal surface M ∞ , in the sense that any sufficiently small normal neighborhood of any smooth compact domain Ω of the limit surface must intersect M n in one or two components that are small normal graphs over Ω for n large. Moreover M ∞ has bounded norm of the second fundamental form, genus at most g and its injectivity radius at the origin is one.…”

“…|A M | denotes the norm of the second fundamental form of M .3. The radius of a Riemannian n-manifold with boundary is the supremum of the intrinsic distances of points in the manifold to its boundary.The next two results are contained in [20], see also [18,19,21,22,23].Theorem 2.2 (Radius Estimates) There exists an R ≥ π such that any H-disk in R 3 has radius less than R/H.…”