Maximum principles for minimal surfaces and for surfaces of continuous mean curvature

1972 ◽  
Vol 128 (3) ◽  
pp. 253-269 ◽  
Author(s):  
Stefan Hildebrandt
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Maximum Principles

scholarly journals LINEAR WEINGARTEN HYPERSURFACES WITH BOUNDED MEAN CURVATURE IN THE HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 653-663 ◽  
Author(s):  
CÍCERO P. AQUINO ◽  
HENRIQUE F. DE LIMA ◽  
MARCO ANTONIO L. VELÁSQUEZ
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Maximum Principles ◽  
Rigidity Result ◽  
The Second Fundamental Form ◽  
Linear Weingarten Hypersurfaces ◽  
Compact Case ◽  
Suitable Restriction

AbstractWe apply appropriate maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. By supposing a suitable restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder, when its scalar curvature is positive, or to a spherical cylinder, when its scalar curvature is negative. Related to the compact case, we also establish a rigidity result.

scholarly journals On minimal surfaces on two-step Carnot groups

2019 ◽  
Vol 485 (4) ◽  
pp. 410-414
Author(s):  
M. B. Karmanova
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Carnot Groups ◽  
Correct Formulation ◽  
Area Functional ◽  
Suitable Notion ◽  
Minimality Conditions

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.  

scholarly journals Self θ-congruent minimal surfaces in ℝ3

2000 ◽  
Vol 69 (2) ◽  
pp. 229-244
Author(s):  
Weihuan Chen ◽  
Yi Fang
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rigid Motions ◽  
Necessary And Sufficient ◽  
Weierstrass Pair

AbstractA minimal surface is a surface with vanishing mean curvature. In this paper we study self θ -congruent minimal surfaces, that is, surfaces which are congruent to their θ-associates under rigid motions in R3 for 0 ≤ θ < 2π. We give necessary and sufficient conditions in terms of its Weierstrass pair for a surface to be self θ-congruent. We also construct some examples and give an application.

scholarly journals Gradient estimates for the constant mean curvature equation in hyperbolic space

2019 ◽  
Vol 150 (6) ◽  
pp. 3216-3230
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gradient Estimates ◽  
Maximum Principles ◽  
Convex Domains ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

AbstractWe establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions.

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