One-sided curvature estimates for H-disks

2014/08/22 by William H. Meeks III, Meeks, William H., Giuseppe Tinaglia +1
Mathematics · #53A10 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53A10

paper · pdf · doi:10.48550/arxiv.1408.5233

Minor corrections. References updated. Format changed

arxiv created 2015/11/03 · arxiv updated 2015/11/04

Abstract

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in ℝ³ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in ℝ³ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in ℝ³ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].

Related

• The geometry of constant mean curvature surfaces in ℝ³
• Limit lamination theorem for H-disks
• On the topology of constant mean curvature surfaces in H2 X R with boundary in a plane
• Constant mean curvature hypersurfaces in \mathbb Hⁿ×\mathbb R with small planar boundary
• Existence of free boundary disks with constant mean curvature in ℝ³