scholarly journals Surfaces of Constant Mean Curvature in Euclidean 3-space Orthogonal to a Plane along its Boundary

2002 ◽  
Vol 74 (1) ◽  
pp. 33-35 ◽  
Author(s):  
PEDRO A. HINOJOSA
Keyword(s):  
Jordan Curve ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Nonzero Mean ◽  
Compact Surfaces ◽  
Planar Jordan Curve

We consider compact surfaces with constant nonzero mean curvature whose boundary is a convex planar Jordan curve. We prove that if such a surface is orthogonal to the plane of the boundary, then it is a hemisphere.

scholarly journals Limit lamination theorems for H-surfaces

2019 ◽  
Vol 2019 (748) ◽  
pp. 269-296 ◽  
Author(s):  
William H. Meeks III ◽  
Giuseppe Tinaglia
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Structure Theorems ◽  
Compact Surfaces ◽  
Finite Genus

AbstractIn this paper we prove some general results for constant mean curvature lamination limits of certain sequences of compact surfacesM_{n}embedded in\mathbb{R}^{3}with constant mean curvatureH_{n}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.

scholarly journals On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 161-191
Author(s):  
Otis Chodosh ◽  
Michael Eichmair
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Results ◽  
Asymptotically Flat ◽  
Weingarten Surfaces ◽  
Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

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