1,038 research outputs found
Covariant Poisson equation with compact Lie algebras
The covariant Poisson equation for Lie algebra-valued mappings defined in
3-dimensional Euclidean space is studied using functional analytic methods.
Weighted covariant Sobolev spaces are defined and used to derive sufficient
conditions for the existence and smoothness of solutions to the covariant
Poisson equation. These conditions require, apart from suitable continuity,
appropriate local integrability of the gauge potentials and global weighted
integrability of the curvature form and the source. The possibility of
nontrivial asymptotic behaviour of a solution is also considered. As a
by-product, weighted covariant generalisations of Sobolev embeddings are
established.Comment: 31 pages, LaTeX2
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
Non-Existence of Positive Stationary Solutions for a Class of Semi-Linear PDEs with Random Coefficients
We consider a so-called random obstacle model for the motion of a
hypersurface through a field of random obstacles, driven by a constant driving
field. The resulting semi-linear parabolic PDE with random coefficients does
not admit a global nonnegative stationary solution, which implies that an
interface that was flat originally cannot get stationary. The absence of global
stationary solutions is shown by proving lower bounds on the growth of
stationary solutions on large domains with Dirichlet boundary conditions.
Difficulties arise because the random lower order part of the equation cannot
be bounded uniformly
A simple expression for the ADM mass
We show by an almost elementary calculation that the ADM mass of an
asymptotically flat space can be computed as a limit involving a rate of change
of area of a closed 2-surface. The result is essentially the same as that given
by Brown and York. We will prove this result in two ways, first by direct
calculation from the original formula as given by Arnowitt, Deser and Misner
and second as a corollary of an earlier result by Brewin for the case of
simplicial spaces.Comment: 9 pages, 1 figur
On a non-isothermal model for nematic liquid crystals
A model describing the evolution of a liquid crystal substance in the nematic
phase is investigated in terms of three basic state variables: the {\it
absolute temperature} \teta, the {\it velocity field} \ub, and the {\it
director field} \bd, representing preferred orientation of molecules in a
neighborhood of any point of a reference domain. The time evolution of the
velocity field is governed by the incompressible Navier-Stokes system, with a
non-isotropic stress tensor depending on the gradients of the velocity and of
the director field \bd, where the transport (viscosity) coefficients vary
with temperature. The dynamics of \bd is described by means of a parabolic
equation of Ginzburg-Landau type, with a suitable penalization term to relax
the constraint |\bd | = 1. The system is supplemented by a heat equation,
where the heat flux is given by a variant of Fourier's law, depending also on
the director field \bd. The proposed model is shown compatible with
\emph{First and Second laws} of thermodynamics, and the existence of
global-in-time weak solutions for the resulting PDE system is established,
without any essential restriction on the size of the data
Time reversal in thermoacoustic tomography - an error estimate
The time reversal method in thermoacoustic tomography is used for
approximating the initial pressure inside a biological object using
measurements of the pressure wave made on a surface surrounding the object.
This article presents error estimates for the time reversal method in the cases
of variable, non-trapping sound speeds.Comment: 16 pages, 6 figures, expanded "Remarks and Conclusions" section,
added one figure, added reference
Global embedding of the Kerr black hole event horizon into hyperbolic 3-space
An explicit global and unique isometric embedding into hyperbolic 3-space,
H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is
given. In particular, this allows the embedding into H^3 of surfaces of
revolution having negative, but finite, Gaussian curvature at smooth fixed
points of the U(1) isometry. As an example, we exhibit the global embedding of
the Kerr-Newman event horizon into H^3, for arbitrary values of the angular
momentum. For this example, considering a quotient of H^3 by the Picard group,
we show that the hyperbolic embedding fits in a fundamental domain of the group
up to a slightly larger value of the angular momentum than the limit for which
a global embedding into Euclidean 3-space is possible. An embedding of the
double-Kerr event horizon is also presented, as an example of an embedding
which cannot be made global.Comment: 16 pages, 13 figure
On geometric problems related to Brown-York and Liu-Yau quasilocal mass
We discuss some geometric problems related to the definitions of quasilocal
mass proposed by Brown-York \cite{BYmass1} \cite{BYmass2} and Liu-Yau
\cite{LY1} \cite{LY2}. Our discussion consists of three parts. In the first
part, we propose a new variational problem on compact manifolds with boundary,
which is motivated by the study of Brown-York mass. We prove that critical
points of this variation problem are exactly static metrics. In the second
part, we derive a derivative formula for the Brown-York mass of a smooth family
of closed 2 dimensional surfaces evolving in an ambient three dimensional
manifold. As an interesting by-product, we are able to write the ADM mass
\cite{ADM61} of an asymptotically flat 3-manifold as the sum of the Brown-York
mass of a coordinate sphere and an integral of the scalar curvature plus
a geometrically constructed function in the asymptotic region outside
. In the third part, we prove that for any closed, spacelike, 2-surface
in the Minkowski space for which the Liu-Yau mass is
defined, if bounds a compact spacelike hypersurface in ,
then the Liu-Yau mass of is strictly positive unless lies on
a hyperplane. We also show that the examples given by \'{O} Murchadha, Szabados
and Tod \cite{MST} are special cases of this result.Comment: 28 page
Differential Geometry of Quantum States, Observables and Evolution
The geometrical description of Quantum Mechanics is reviewed and proposed as
an alternative picture to the standard ones. The basic notions of observables,
states, evolution and composition of systems are analised from this
perspective, the relevant geometrical structures and their associated algebraic
properties are highlighted, and the Qubit example is thoroughly discussed.Comment: 20 pages, comments are welcome
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