47 research outputs found
A volumetric Penrose inequality for conformally flat manifolds
We consider asymptotically flat Riemannian manifolds with nonnegative scalar
curvature that are conformal to , and so that
their boundary is a minimal hypersurface. (Here, is open
bounded with smooth mean-convex boundary.) We prove that the ADM mass of any
such manifold is bounded below by , where is the
Euclidean volume of and is the volume of the Euclidean
unit -ball. This gives a partial proof to a conjecture of Bray and Iga
\cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.Comment: 7 page
Asymptotically hyperbolic manifolds with small mass
For asymptotically hyperbolic manifolds of dimension with scalar
curvature at least equal to the conjectured positive mass theorem
states that the mass is non-negative, and vanishes only if the manifold is
isometric to hyperbolic space. In this paper we study asymptotically hyperbolic
manifolds which are also conformally hyperbolic outside a ball of fixed radius,
and for which the positive mass theorem holds. For such manifolds we show that
the conformal factor tends to one as the mass tends to zero
Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds
This article is the sequel to our previous paper [LS] dealing with the
near-equality case of the Positive Mass Theorem. We study the near-equality
case of the Penrose Inequality for the class of complete asymptotically flat
rotationally symmetric Riemannian manifolds with nonnegative scalar curvature
whose boundaries are outermost minimal hypersurfaces. Specifically, we prove
that if the Penrose Inequality is sufficiently close to being an equality on
one of these manifolds, then it must be close to a Schwarzschild space with an
appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz
Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a
result for Intrinsic Flat Distance, which is a more appropriate distance for
more general near-equality results, as discussed in [LS]Comment: 19 pages, 2 figure
Blowup of Jang's equation at outermost marginally trapped surfaces
The aim of this paper is to collect some facts about the blowup of Jang's
equation. First, we discuss how to construct solutions that blow up at an
outermost MOTS. Second, we exclude the possibility that there are extra blowup
surfaces in data sets with non-positive mean curvature. Then we investigate the
rate of convergence of the blowup to a cylinder near a strictly stable MOTS and
show exponential convergence near a strictly stable MOTS.Comment: 15 pages. This revision corrects some typo
On a Localized Riemannian Penrose Inequality
Consider a compact, orientable, three dimensional Riemannian manifold with
boundary with nonnegative scalar curvature. Suppose its boundary is the
disjoint union of two pieces: the horizon boundary and the outer boundary,
where the horizon boundary consists of the unique closed minimal surfaces in
the manifold and the outer boundary is metrically a round sphere. We obtain an
inequality relating the area of the horizon boundary to the area and the total
mean curvature of the outer boundary. Such a manifold may be thought as a
region, surrounding the outermost apparent horizons of black holes, in a
time-symmetric slice of a space-time in the context of general relativity. The
inequality we establish has close ties with the Riemannian Penrose Inequality,
proved by Huisken and Ilmanen, and by Bray.Comment: 16 page
Some Curvature Problems in Semi-Riemannian Geometry
In this survey article we review several results on the curvature of
semi-Riemannian metrics which are motivated by the positive mass theorem. The
main themes are estimates of the Riemann tensor of an asymptotically flat
manifold and the construction of Lorentzian metrics which satisfy the dominant
energy condition.Comment: 25 pages, LaTeX, 4 figure
The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight
We extend the Jang equation proof of the positive energy theorem due to R.
Schoen and S.-T. Yau from dimension to dimensions . This
requires us to address several technical difficulties that are not present when
. The regularity and decay assumptions for the initial data sets to which
our argument applies are weaker than those of R. Schoen and S.-T. Yau. In
recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a
different proof of the full positive mass theorem in dimensions .
We pointed out that this theorem can alternatively be derived from our density
argument and the positive energy theorem of the present paper.Comment: All comments welcome! Final version to appear in Comm. Math. Phy
Adaptive cluster expansion for the inverse Ising problem: convergence, algorithm and tests
We present a procedure to solve the inverse Ising problem, that is to find
the interactions between a set of binary variables from the measure of their
equilibrium correlations. The method consists in constructing and selecting
specific clusters of variables, based on their contributions to the
cross-entropy of the Ising model. Small contributions are discarded to avoid
overfitting and to make the computation tractable. The properties of the
cluster expansion and its performances on synthetic data are studied. To make
the implementation easier we give the pseudo-code of the algorithm.Comment: Paper submitted to Journal of Statistical Physic
A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory
In this paper we present a proof of a mathematical version of the strong
cosmic censor conjecture attributed to Geroch-Horowitz and Penrose but
formulated explicitly by Wald. The proof is based on the existence of
future-inextendible causal curves in causal pasts of events on the future
Cauchy horizon in a non-globally hyperbolic space-time. By examining explicit
non-globally hyperbolic space-times we find that in case of several physically
relevant solutions these future-inextendible curves have in fact infinite
length. This way we recognize a close relationship between asymptotically flat
or anti-de Sitter, physically relevant extendible space-times and the so-called
Malament-Hogarth space-times which play a central role in recent investigations
in the theory of "gravitational computers". This motivates us to exhibit a more
sharp, more geometric formulation of the strong cosmic censor conjecture,
namely "all physically relevant, asymptotically flat or anti-de Sitter but
non-globally hyperbolic space-times are Malament-Hogarth ones".
Our observations may indicate a natural but hidden connection between the
strong cosmic censorship scenario and the Church-Turing thesis revealing an
unexpected conceptual depth beneath both conjectures.Comment: 16pp, LaTeX, no figures. Final published versio