165 research outputs found
A rigidity theorem for nonvacuum initial data
In this note we prove a theorem on non-vacuum initial data for general
relativity. The result presents a ``rigidity phenomenon'' for the extrinsic
curvature, caused by the non-positive scalar curvature.
More precisely, we state that in the case of asymptotically flat non-vacuum
initial data if the metric has everywhere non-positive scalar curvature then
the extrinsic curvature cannot be compactly supported.Comment: This is an extended and published version: LaTex, 10 pages, no
figure
Minimal immersions of closed surfaces in hyperbolic three-manifolds
We study minimal immersions of closed surfaces (of genus ) in
hyperbolic 3-manifolds, with prescribed data , where
is a conformal structure on a topological surface , and is a holomorphic quadratic differential on the surface . We
show that, for each for some , depending only on
, there are at least two minimal immersions of closed surface
of prescribed second fundamental form in the conformal structure
. Moreover, for sufficiently large, there exists no such minimal
immersion. Asymptotically, as , the principal curvatures of one
minimal immersion tend to zero, while the intrinsic curvatures of the other
blow up in magnitude.Comment: 16 page
A generalization of Hawking's black hole topology theorem to higher dimensions
Hawking's theorem on the topology of black holes asserts that cross sections
of the event horizon in 4-dimensional asymptotically flat stationary black hole
spacetimes obeying the dominant energy condition are topologically 2-spheres.
This conclusion extends to outer apparent horizons in spacetimes that are not
necessarily stationary. In this paper we obtain a natural generalization of
Hawking's results to higher dimensions by showing that cross sections of the
event horizon (in the stationary case) and outer apparent horizons (in the
general case) are of positive Yamabe type, i.e., admit metrics of positive
scalar curvature. This implies many well-known restrictions on the topology,
and is consistent with recent examples of five dimensional stationary black
hole spacetimes with horizon topology . The proof is inspired
by previous work of Schoen and Yau on the existence of solutions to the Jang
equation (but does not make direct use of that equation).Comment: 8 pages, latex2e, references updated, minor corrections, to appear in
Communications in Mathematical Physic
A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
We establish new existence and non-existence results for positive solutions
of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This
equation arises from the Hamiltonian constraint equation for the
Einstein-scalar field system in general relativity. Our analysis introduces
variational techniques, in the form of the mountain pass lemma, to the analysis
of the Hamiltonian constraint equation, which has been previously studied by
other methods.Comment: 15 page
Metrics with Prescribed Ricci Curvature near the Boundary of a Manifold
Suppose is a manifold with boundary. Choose a point . We
investigate the prescribed Ricci curvature equation \Ric(G)=T in a
neighborhood of under natural boundary conditions. The unknown here is
a Riemannian metric. The letter in the right-hand side denotes a
(0,2)-tensor. Our main theorems address the questions of the existence and the
uniqueness of solutions. We explain, among other things, how these theorems may
be used to study rotationally symmetric metrics near the boundary of a solid
torus . The paper concludes with a brief discussion of the Einstein
equation on .Comment: 13 page
Rotational symmetry of self-similar solutions to the Ricci flow
Let (M,g) be a three-dimensional steady gradient Ricci soliton which is
non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the
Bryant soliton up to scaling. This solves a problem mentioned in Perelman's
first paper.Comment: Final version, to appear in Invent. Mat
Initial Data for General Relativity Containing a Marginally Outer Trapped Torus
Asymptotically flat, time-symmetric, axially symmetric and conformally flat
initial data for vacuum general relativity are studied numerically on
with the interior of a standard torus cut out. By the choice of boundary
condition the torus is marginally outer trapped, and thus a surface of minimal
area. Apart from pure scaling the standard tori are parameterized by a radius
, where corresponds to the limit where the boundary torus
degenerates to a circle and to a torus that touches the axis of symmetry.
Noting that these tori are the orbits of a conformal isometry
allows for a simple scheme to solve the constraint, involving numerical
solution of only ordinary differential equations.The tori are unstable minimal
surfaces (i.e. only saddle points of the area functional) and thus can not be
apparent horizons, but are always surrounded by an apparent horizon of
spherical topology, which is analyzed in the context of the hoop conjecture and
isoperimetric inequality for black holes.Comment: 12 pages, REVTeX 3.0, also available (with additional pictures and
numerical data) from http://doppler.thp.univie.ac.at/~shusa/gr.htm
On the topology and area of higher dimensional black holes
Over the past decade there has been an increasing interest in the study of
black holes, and related objects, in higher (and lower) dimensions, motivated
to a large extent by developments in string theory. The aim of the present
paper is to obtain higher dimensional analogues of some well known results for
black holes in 3+1 dimensions. More precisely, we obtain extensions to higher
dimensions of Hawking's black hole topology theorem for asymptotically flat
() black hole spacetimes, and Gibbons' and Woolgar's genus
dependent, lower entropy bound for topological black holes in asymptotically
locally anti-de Sitter () spacetimes. In higher dimensions the genus
is replaced by the so-called -constant, or Yamabe invariant, which is a
fundamental topological invariant of smooth compact manifolds.Comment: 15 pages, Latex2e; typos corrected, a convention clarified, resulting
in the simplification of certain formulas, other improvement
About curvature, conformal metrics and warped products
We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds and furnished with metrics of
the form and, in particular, of the type , where are smooth
functions and is a real parameter. We obtain suitable expressions for the
Ricci tensor and scalar curvature of such products that allow us to establish
results about the existence of Einstein or constant scalar curvature structures
in these categories. If is Riemannian, the latter question involves
nonlinear elliptic partial differential equations with concave-convex
nonlinearities and singular partial differential equations of the
Lichnerowicz-York type among others.Comment: 32 pages, 3 figure
A spinorial energy functional: critical points and gradient flow
On the universal bundle of unit spinors we study a natural energy functional
whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi})
consisting of a Ricci-flat Riemannian metric g together with a parallel
g-spinor {\phi}. We investigate the basic properties of this functional and
study its negative gradient flow, the so-called spinor flow. In particular, we
prove short-time existence and uniqueness for this flow.Comment: Small changes, final versio
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