91 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
Exchange Operator Formalism for Integrable Systems of Particles
We formulate one dimensional many-body integrable systems in terms of a new
set of phase space variables involving exchange operators. The hamiltonian in
these variables assumes a decoupled form. This greatly simplifies the
derivation of the conserved charges and the proof of their commutativity at the
quantum level.Comment: 8 page
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
Four conjectures in Nonlinear Analysis
In this chapter, I formulate four challenging conjectures in Nonlinear
Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a
conjecture on an eigenvalue problem; a conjecture on a non-local problem; a
conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010,
arXiv:1409.5919, arXiv:1612.0819
A supercritical elliptic problem in a cylindrical shell
We consider the problem where
, and . Let
if and if
or . We show that is the true critical exponent
for this problem, and that there exist nontrivial solutions if
but there are no such solutions if
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
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
The Cauchy Problem for the Einstein Equations
Various aspects of the Cauchy problem for the Einstein equations are
surveyed, with the emphasis on local solutions of the evolution equations.
Particular attention is payed to giving a clear explanation of conceptual
issues which arise in this context. The question of producing reduced systems
of equations which are hyperbolic is examined in detail and some new results on
that subject are presented. Relevant background from the theory of partial
differential equations is also explained at some lengthComment: 98 page
Magnetic vortex filament flows
We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.Ministerio de Educación y CienciaFondo Europeo de Desarrollo RegionalJunta de Andalucí
Local and global behaviour of nonlinear equations with natural growth terms
This paper concerns a study of the pointwise behaviour of positive solutions
to certain quasi-linear elliptic equations with natural growth terms, under
minimal regularity assumptions on the underlying coefficients. Our primary
results consist of optimal pointwise estimates for positive solutions of such
equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto
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