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.

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 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.

The Kerr solution

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Kerr Black Hole ◽  
Geodesic Equation ◽  
Einstein Equations ◽  
Kerr Metric ◽  
Kerr Solution ◽  
Axially Symmetric ◽  
Observable Universe ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
The Many

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

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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