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 On pinching theorems for compact pseudo-umbilical submanifold

2012 ◽  
Vol 45 (3) ◽  
pp. 645-654
Author(s):  
Jing Mao ◽  
Shaodong Qin
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Parallel Mean Curvature

AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

scholarly journals Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Sebastian Heller ◽  
Nick Schmitt
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Weierstrass Representation ◽  
Geometric Flow ◽  
Constant Mean Curvature Surfaces ◽  
Cmc Surfaces

AbstractWe describe the construction of CMC surfaces with symmetries in $\mathbb {S}^{3}$ S 3 and $\mathbb {R}^{3}$ ℝ 3 using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

Causal geometry of rough Einstein CMCSH spacetime

2014 ◽  
Vol 11 (03) ◽  
pp. 563-601 ◽  
Author(s):  
Qian Wang
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Strichartz Estimate ◽  
Einstein Metric ◽  
Well Posedness ◽  
Spatial Harmonic ◽  
Integral Control ◽  
Field Approach ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum

This is the second (and last) part of a series in which we consider very rough solutions to Cauchy problem for the Einstein vacuum equations in constant mean curvature and spatial harmonic (CMCSH) gauge, and we obtain a local well-posedness result in Hs with s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough Einstein metric g, we manage to implement the commuting vector field approach and prove a Strichartz estimate for the geometric wave equation □g ϕ = 0 in a direct manner. This direct treatment would not work without gaining sufficient regularity on the background geometry. In this paper, we analyze the geometry of null hypersurfaces in rough Einstein spacetimes in terms of Hs data. We provide an integral control on the spatial supremum of the connection coefficients [Formula: see text], ζ, which is crucially tied to the Strichartz estimates established in the first part.

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.

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.

Constant-mean-curvature slicing of the Schwarzschild-de Sitter space-time

1991 ◽  
Vol 44 (4) ◽  
pp. 1326-1329 ◽  
Author(s):  
Ken-ichi Nakao ◽  
Kei-ichi Maeda ◽  
Takashi Nakamura ◽  
Ken-ichi Oohara
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
De Sitter Space ◽  
Space Time ◽  
De Sitter

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 A CLASS OF COMPACT POINCARÉ–EINSTEIN MANIFOLDS: PROPERTIES AND CONSTRUCTION

2010 ◽  
Vol 12 (04) ◽  
pp. 629-659 ◽  
Author(s):  
A. ROD GOVER ◽  
FELIPE LEITNER
Keyword(s):  
General Theory ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Explicit Construction ◽  
Einstein Metric ◽  
Dense Subspace ◽  
Einstein Manifolds ◽  
Construction Principle ◽  
Conformal Structures

We develop a geometric and explicit construction principle that generates classes of Poincaré–Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalization of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincaré–Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classification results which show that the construction essentially exhausts a class of almost Einstein (and Poincaré–Einstein) manifolds. We develop the general theory of fixed conformal structures admitting multiple compatible almost Einstein structures. We also show that, in a class of cases, these are canonically related to a family of constant mean curvature totally umbillic embedded hypersurfaces.

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