scholarly journals DISCRETE LINEAR WEINGARTEN SURFACES

2017 ◽  
Vol 231 ◽  
pp. 55-88 ◽  
Author(s):  
F. BURSTALL ◽  
U. HERTRICH-JEROMIN ◽  
W. ROSSMAN
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Discrete Analogue ◽  
Space Forms ◽  
Weingarten Surfaces ◽  
Lawson Correspondence ◽  
Geometric Deformation

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

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.

scholarly journals Constant mean curvature hypersurfaces with constant angle in semi-Riemannian space forms

2016 ◽  
Vol 49 ◽  
pp. 473-495 ◽  
Author(s):  
Matias Navarro ◽  
Gabriel Ruiz-Hernández ◽  
Didier A. Solis
Keyword(s):  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Forms ◽  
Constant Mean Curvature Hypersurfaces ◽  
Constant Angle ◽  
Riemannian Space Forms

scholarly journals A note on Weingarten hypersurfaces in the warped product manifold

2014 ◽  
Vol 25 (14) ◽  
pp. 1450121 ◽  
Author(s):  
Haizhong Li ◽  
Yong Wei ◽  
Changwei Xiong
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Higher Order ◽  
Product Manifold ◽  
Weingarten Surfaces ◽  
Warped Product Manifold ◽  
Higher Order Mean Curvature ◽  
Special Case ◽  
Differential Geom

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

scholarly journals Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

scholarly journals Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

2007 ◽  
Vol 75 (3) ◽  
pp. 563-581 ◽  
Author(s):  
N. Schmitt ◽  
M. Kilian ◽  
S.-P. Kobayashi ◽  
W. Rossman
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Space ◽  
Space Forms ◽  
3 Dimensional

scholarly journals A deformation of tori with constant mean curvature in ${\bf R}\sp 3$ to those in other space forms

1992 ◽  
Vol 330 (2) ◽  
pp. 845-857
Author(s):  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Forms

Biharmonic hypersurfaces in pseudo-Riemannian space forms

2016 ◽  
Vol 13 (07) ◽  
pp. 1650094 ◽  
Author(s):  
Dan Yang ◽  
Yu Fu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Constant Sectional Curvature ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

A Deformation of Tori with Constant Mean Curvature in ℝ 3 to Those in Other Space Forms

1992 ◽  
Vol 330 (2) ◽  
pp. 845 ◽  
Author(s):  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Forms
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