One-sided curvature estimates for H-disks

Limit lamination theorem for H-disks

“…Proof The proof of this claim follows the arguments in the beginning of the proof of Case A of Proposition 3.1 in [24]. Note that in Proposition 3.1 in [24] M n is denoted by (n). For the sake of completeness we outline the main arguments in the proof of this fact.…”

“…Note that if Case A holds, we will prove item A of Theorem 1.1 holds and if Case B holds, we will prove that item B of Theorem 1.1 holds. Some of the arguments in the proofs of Cases A and B listed above are borrowed from arguments appearing in the proofs of similar Cases A and B in the proof of Proposition 3.1 in [24], where all the details are given.…”

“…Consequently, the reader may wish to consult the proof of Proposition 3.1 in [24] before continuing to read what follows.…”

“…In this paper we apply results in [22][23][24] to obtain (after passing to a subsequence) minimal lamination limits for any sequence of compact disks M n embedded in R 3 with constant mean curvature H n , when the boundaries of these disks tend to infinity; see Theorem 1.1 below. This theorem is inspired by, and generalizes to the non-zero constant mean curvature setting, Theorem 0.1 by Colding and Minicozzi [7] and it is related to work in [3,14,15,26].…”

“…The proofs of the results described in this paper depend in an essential manner on the existence of extrinsic curvature estimates for disks embedded in R 3 of non-zero constant mean curvature that appear in [23], as well as on a key extrinsic one-sided curvature estimate obtained in [24] and a weak cord arc result derived in [22]; these results from [22][23][24] are described in Sect. 2.…”