# Generalized Inverse Mean Curvature Flows in Spacetime

@article{Bray2007GeneralizedIM,
title={Generalized Inverse Mean Curvature Flows in Spacetime},
author={Hubert L. Bray and Sean A. Hayward and Marc Mars and Walter Simon},
journal={Communications in Mathematical Physics},
year={2007},
volume={272},
pages={119-138}
}
• Published 7 March 2006
• Mathematics
• Communications in Mathematical Physics
Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the…
Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass
• Physics, Mathematics
• 2015
We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime
Uniformly Area Expanding Flows in Spacetimes
The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse
Spacelike submanifolds, their umbilical properties and applications to gravitational physics.
We give a characterization theorem for umbilical spacelike submanifolds of arbitrary dimension and co-dimension immersed in a semi-Riemannian manifold. Letting the codimension arbitrary implies that
Foliations of null hypersurfaces and the Penrose inequality
Starting point and motivation for this thesis rest upon a study of the Penrose inequality in the null case for asymptotically flat vacuum spacetimes (M, g) under the assumption that the past light
Solitons for the inverse mean curvature flow
• Mathematics
• 2015
We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and
Umbilical Properties of Spacelike Co-dimension Two Submanifolds
• Mathematics
• 2016
For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of total shear tensor and shear operators as the trace-free part of the corresponding second fundamental form and
Deformation of codimension-2 surfaces and horizon thermodynamics
The deformation equation of a spacelike submanifold with an arbitrary codimension is given by a general construction without using local frames. In the case of codimension-1, this equation reduces to
Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II
• Mathematics, Physics
• 2016
In this sequel paper, we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the
On the construction of Riemannian three-spaces with smooth generalized inverse mean curvature flows
Choose a smooth three-dimensional manifold $\Sigma$ that is smoothly foliated by topological two-spheres, and also a smooth flow on $\Sigma$ such that the integral curves of it intersect the leaves
The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces
• Mathematics
• 2015
In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $\Omega$ satisfying a weak notion of asymptotic flatness. The foliations are not

## References

SHOWING 1-10 OF 29 REFERENCES
PROOF OF THE RIEMANNIAN PENROSE INEQUALITY USING THE POSITIVE MASS THEOREM
We prove the Riemannian Penrose Conjecture, an important case of a con- jecture (41) made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of
On the Penrose inequality for general horizons.
• Mathematics
Physical review letters
• 2002
For asymptotically flat initial data of Einstein's equations satisfying an energy condition, we show that the Penrose inequality holds between the Arnowitt-Deser-Misner mass and the area of an
The inverse mean curvature flow and the Riemannian Penrose Inequality
• Mathematics
• 2001
Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m
Some comments on gravitational entropy and the inverse mean curvature flow
The Geroch-Wald-Jang-Huisken-Ilmanen approach to the positive energy problem may be extended to give a negative lower bound for the mass of asymptotically anti-de Sitter spacetimes containing
The Penrose Inequality
• Mathematics
• 2004
In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area A should be at least $$\sqrt {A/16\pi}$$. An important
The Global Nonlinear Stability of the Minkowski Space.
• Mathematics
• 1994
The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to
Trapped surfaces and the Penrose inequality in spherically symmetric geometries.
• Mathematics
Physical review. D, Particles and fields
• 1994
It is demonstrated that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter, and a modification of the penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse.
Spatial and null infinity via advanced and retarded conformal factors
A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. Generalizing examples
Quasi-localization of Bondi-Sachs energy loss
A formula is given for the variation of the Hawking energy along any one-parameter family of compact spatial 2-surfaces. A surface for which one null expansion is positive and the other negative has
Riemannian geometry and geometric analysis
* Established textbook * Continues to lead its readers to some of the hottest topics of contemporary mathematical research This established reference work continues to lead its readers to some of