Plateau's problem for surfaces of constant mean curvature from a global point of view

1983 ◽  
Vol 43 (2-3) ◽  
pp. 191-217 ◽  
Author(s):  
Michael S�llner
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Point Of View ◽  
Plateau’S Problem ◽  
Plateau's Problem

scholarly journals Triply periodic minimal and constant mean curvature surfaces

2012 ◽  
Vol 2 (5) ◽  
pp. 582-588 ◽  
Author(s):  
Karsten Grosse-Brauckmann
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Point Of View ◽  
Constant Mean Curvature Surfaces ◽  
Mathematical Point ◽  
Periodic Surfaces ◽  
Triply Periodic ◽  
General Readership

We want to summarize some established results on periodic surfaces which are minimal or have constant mean curvature, along with some recent results. We will do this from a mathematical point of view with a general readership in mind.

scholarly journals On Bernstein–Heinz–Chern–Flanders inequalities

2008 ◽  
Vol 144 (2) ◽  
pp. 457-464 ◽  
Author(s):  
J. L. M. BARBOSA ◽  
G. P. BESSA ◽  
J. F. MONTENEGRO
Keyword(s):  
Lower Bounds ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Point Of View ◽  
Constant Mean Curvature Surfaces ◽  
3 Dimensional ◽  
Codimension One ◽  
Simple Curves

AbstractWe give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.

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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