Limit lamination theorem for H-disks

Abstract

We describe the lamination limits of sequences of compact disks $$M_n$$ M n embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature $$H_n$$ 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 (Ann Math 160:573–615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi (Ann Math 167:211–243, 2008) for minimal disks.

Limit lamination theorems for H-surfaces

“…Corollary 2.6 (Corollary 4.6 in [30]). There exist constants ε < 1, C > 1 such that the following holds.…”

“…2 ). For applications here, we will also need the closely related chord-arc result below, that is Theorem 1.2 in [30]. Theorem 2.9 (Chord arc property for H-disks).…”

“…Since in the proofs of Theorems 1.3, 1.4 and 1.5 we will frequently refer to parts of the statement of the Limit Lamination Theorem for H-disks, namely Theorem 1.1 in [30], we state it below for more direct referencing.…”

“…(b) There exists compact subdomains [30], after replacing the surfaces M n by a subsequence and composing them by a rotation of R 3 that fixes the origin and so that the planes of the limit foliation are horizontal, then, as n tends to infinity, ∆ n has the structure of a double spiral staircase, in the following sense:…”

“…In this paper we apply results in [28,29,30,31] to obtain (after passing to a subsequence) constant mean curvature lamination limits for sequences of compact surfaces M n embedded in R 3 with constant mean curvature H n and fixed finite genus, when the boundaries of these surfaces tend to infinity in R 3 . These lamination limit results are inspired by and generalize to the non-zero constant mean curvature setting similar structure theorems by Colding and Minicozzi in [6,8] in the case of embedded minimal surfaces; also see some closely related work of Meeks, Perez and Ros in [19,21] in the minimal setting.…”