275 research outputs found
Geometric Analysis and General Relativity
This article discusses methods of geometric analysis in general relativity,
with special focus on the role of "critical surfaces" such as minimal surfaces,
marginal surface, maximal surfaces and null surfaces.Comment: to appear in Elsevier Encyclopedia of Mathematical Physics, 200
Weak Continuity and Compactness for Nonlinear Partial Differential Equations
We present several examples of fundamental problems involving weak continuity
and compactness for nonlinear partial differential equations, in which
compensated compactness and related ideas have played a significant role. We
first focus on the compactness and convergence of vanishing viscosity solutions
for nonlinear hyperbolic conservation laws, including the inviscid limit from
the Navier-Stokes equations to the Euler equations for homentropy flow, the
vanishing viscosity method to construct the global spherically symmetric
solutions to the multidimensional compressible Euler equations, and the
sonic-subsonic limit of solutions of the full Euler equations for
multidimensional steady compressible fluids. We then analyze the weak
continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding
isometric embeddings in differential geometry. Further references are also
provided for some recent developments on the weak continuity and compactness
for nonlinear partial differential equations.Comment: 29 page
Universality in mean curvature flow neckpinches
We study noncompact surfaces evolving by mean curvature flow. Without any
symmetry assumptions, we prove that any solution that is -close at some
time to a standard neck will develop a neckpinch singularity in finite time,
will become asymptotically rotationally symmetric in a space-time neighborhood
of its singular set, and will have a unique tangent flow.Comment: More references added, typos correcte
Neckpinch dynamics for asymmetric surfaces evolving by mean curvature flow
We study surfaces evolving by mean curvature flow (MCF). For an open set of
initial data that are -close to round, but without assuming rotational
symmetry or positive mean curvature, we show that MCF solutions become singular
in finite time by forming neckpinches, and we obtain detailed asymptotics of
that singularity formation. Our results show in a precise way that MCF
solutions become asymptotically rotationally symmetric near a neckpinch
singularity.Comment: This revision corrects minor but potentially confusing misprints in
Section
Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes
The present work extends our short communication Phys. Rev. Lett. 95, 111102
(2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth
spacetime we define stability with respect to variations along arbitrary
vectors v normal to the MOTS. After giving some introductory material about
linear non self-adjoint elliptic operators, we introduce the stability operator
L_v and we characterize stable MOTS in terms of sign conditions on the
principal eigenvalue of L_v. The main result shows that given a strictly stable
MOTS S contained in one leaf of a given reference foliation in a spacetime,
there is an open marginally outer trapped tube (MOTT), adapted to the reference
foliation, which contains S. We give conditions under which the MOTT can be
completed. Finally, we show that under standard energy conditions on the
spacetime, the MOTT must be either locally achronal, spacelike or null.Comment: 33 pages, no figures, typos corrected, minor changes in presentatio
Foliations for Quasi-Fuchsian 3-Manifolds
In this paper, we prove that if a quasi-Fuchsian 3-manifold contains a
minimal surface whose principle curvature is less than 1, then it admits a
foliation such that each leaf is a surface of constant mean curvature. The key
method that we use here is volume preserving mean curvature flow.Comment: 22 pages, 2 figure
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