Lie geometry of linear Weingarten surfaces

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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Gap theorems for linear Weingarten surfaces

ANNALI DELL UNIVERSITA DI FERRARA ◽

10.1007/s11565-020-00336-4 ◽

2020 ◽

Vol 66 (1) ◽

pp. 89-98

Author(s):

Henrique F. De Lima ◽

Fábio R. Dos Santos

Keyword(s):

Weingarten Surfaces ◽

Gap Theorems

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Channel Linear Weingarten Surfaces

Journal of Geometry and Symmetry in Physics ◽

10.7546/jgsp-40-2015-25-33 ◽

2015 ◽

Vol 40 ◽

pp. 25-33 ◽

Cited By ~ 2

Author(s):

Udo Hertrich-Jeromin ◽

◽

Klara Mundilova ◽

Ekkehard-Heinrich Tjaden

Keyword(s):

Weingarten Surfaces

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On the structure of certain Weingarten surfaces with boundary a circle

Annales de la faculté des sciences de Toulouse Mathématiques ◽

10.5802/afst.863 ◽

1997 ◽

Vol 6 (2) ◽

pp. 243-255 ◽

Cited By ~ 7

Author(s):

Fabiano Gustavo Braga Brito ◽

Ricardo Sa Earp

Keyword(s):

Weingarten Surfaces

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Weingarten surfaces and sine-Gordon equation

Science in China Series A Mathematics ◽

10.1007/bf03182362 ◽

1997 ◽

Vol 40 (10) ◽

pp. 1028-1035 ◽

Cited By ~ 1

Author(s):

Weihuan Chen ◽

Haizhong Li

Keyword(s):

Gordon Equation ◽

Sine Gordon Equation ◽

Weingarten Surfaces

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Isothermic surfaces in sphere geometries as Moutard nets

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1902 ◽

2007 ◽

Vol 463 (2088) ◽

pp. 3171-3193 ◽

Cited By ~ 6

Author(s):

Alexander I Bobenko ◽

Yuri B Suris

Keyword(s):

Dimensional Space ◽

Three Dimensional ◽

Gauss Map ◽

Point Property ◽

Property A ◽

Lie Geometry ◽

Möbius Geometry ◽

Isothermic Surfaces ◽

Three Dimensional Space

We give an elaborated treatment of discrete isothermic surfaces and their analogues in different geometries (projective, Möbius, Laguerre and Lie). We find the core of the theory to be a novel characterization of discrete isothermic nets as Moutard nets. The latter are characterized by the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equation. Moutard nets admit also a projective geometric characterization as nets with planar faces with a five-point property: a vertex and its four diagonal neighbours span a three-dimensional space. Restricting the projective theory to quadrics, we obtain Moutard nets in sphere geometries. In particular, Moutard nets in Möbius geometry are shown to coincide with discrete isothermic nets. The five-point property, in this particular case, states that a vertex and its four diagonal neighbours lie on a common sphere, which is a novel characterization of discrete isothermic surfaces. Discrete Laguerre isothermic surfaces are defined through the corresponding five-plane property, which requires that a plane and its four diagonal neighbours share a common touching sphere. Equivalently, Laguerre isothermic surfaces are characterized by having an isothermic Gauss map. S-isothermic surfaces as an instance of Moutard nets in Lie geometry are also discussed.

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