scholarly journals Conformal variation of a hypersurface of constant mean curvature in an einstein space

1982 ◽  
Vol 33 (2) ◽  
pp. 229-234 ◽  
Author(s):  
Geoffrey Howard Smith
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Einstein Space ◽  
Totally Umbilical ◽  
Orientable Manifold ◽  
Variation Vector

AbstractLet a compact orientable manifold be immersed as a hypersurface of constant mean curvature in an Einstein space. It is shown that the immersion is totally umbilical if and only if there exists a conformal variation of the immersion whose variation vector is nowhere tangential to the hypersurface.

Notes on Hypersurfaces in a Riemannian Manifold

1967 ◽  
Vol 19 ◽  
pp. 439-446 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Killing Vector ◽  
Einstein Space ◽  
Inner Product ◽  
Killing Vector Field ◽  
Parameter Group ◽  
Fixed Sign ◽  
Orientable Hypersurface

H. Liebmann (3) and W. Süss (7) provedTheorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere.Y. Katsurada (1; 2) gave the following generalization.Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical.

Non-immersibility of a space form as a totally umbilical hypersurface

1988 ◽  
Vol 109 (1-2) ◽  
pp. 17-21
Author(s):  
Masafumi Okumura ◽  
Hiroshi Takahashi
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Ricci Tensor ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Einstein Manifold ◽  
Totally Umbilical ◽  
Ambient Manifold

SynopsisSuppose that a space form is immersed into another Riemannian manifold as a totally umbilical hypersurface with constant mean curvature. Then, in the ambient manifold, the lengthof the curvature tensor, that of the Ricci tensor and the scalar curvature must satisfy an inequality. In this paper the authors proved the inequality. As applications of the inequality, some immersibility problems are investigated. For example, it is proved that if a space form is immersed in an Einstein manifold as a totally umbilical hypersurface, then the Einstein manifold has constant sectional curvature along the hypersurface. Moreover, it is proved that a space form cannot be immersed into some Kaehlerian manifolds as a totally umbilical hypersurface with constant mean curvature.

Einstein's hyperbolic geometric flow

2014 ◽  
Vol 11 (02) ◽  
pp. 249-267 ◽  
Author(s):  
De-Xing Kong ◽  
Jinhua Wang
Keyword(s):  
Global Existence ◽  
Wave Phenomenon ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Time ◽  
Einstein Metric ◽  
Geometric Flow ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold

We investigate the Einstein's hyperbolic geometric flow, which provides a natural tool to deform the shape of a manifold and to understand the wave character of metrics, the wave phenomenon of the curvature for evolutionary manifolds. For an initial manifold equipped with an Einstein metric and assumed to be a totally umbilical submanifold in the induced space-time, we prove that, along the Einstein's hyperbolic geometric flow, the metric is Einstein if and only if the corresponding manifold is a totally umbilical hypersurface in the induced space-time. For an initial manifold which is equipped with an Einstein metric, assumed to be a totally umbilical submanifold with constant mean curvature in the induced space-time, we prove that, along the Einstein's hyperbolic geometric flow, the metric remains an Einstein metric, and the corresponding manifold is a totally umbilical hypersurface in the induced space-time. Moreover, the global existence and blowup phenomenon of the constructed metric is also investigated here.

scholarly journals Totally umbilical hypersurfaces of Spinc manifolds carrying special spinor fields

2020 ◽  
Vol 31 (12) ◽  
pp. 2050100
Author(s):  
Nadine Große ◽  
Roger Nakad
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Einstein Manifold ◽  
Constant Sectional Curvature ◽  
Killing Spinor ◽  
Totally Umbilical ◽  
Totally Umbilical Hypersurfaces ◽  
Spinor Fields

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin[Formula: see text] manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin[Formula: see text] case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to [Formula: see text] is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian [Formula: see text] manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

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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