610 research outputs found
Causal Geometry of Einstein-Vacuum Spacetimes with Finite Curvature Flux
One of the central difficulties of settling the -bounded curvature
conjecture for the Einstein -Vacuum equations is to be able to control the
causal structure of spacetimes with such limited regularity.
In this paper we show how to circumvent this difficulty by showing that the
geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled
in terms of initial data and the total curvature flux through the hypersurface.Comment: The previous version has been corrected for minor errors and
expositio
Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions
We show that in dimensions that one has global regularity for the
Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical
Sobolev norm of the initial data is
sufficiently small. These results are analogous to those recently obtained for
the high-dimensional wave map equation but unlike the wave map equation, the
Coulomb gauge non-linearity cannot be iterated away directly. We shall use a
different approach, proving Strichartz estimates for the covariant wave
equation. This in turn will be achieved by use of Littlewood-Paley multipliers,
and a global parametrix for the covariant wave equation constructed using a
truncated, microlocalized Cronstrom gauge.Comment: 49 pages, no pictures, to appear, CMP. A minor problem with a Fourier
angular projection causing a certain phase function to no longer be real has
now been fixe
A Generalized Representation Formula for Systems of Tensor Wave Equations
In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman
and Rodnianski to systems of tensor wave equations with additional first-order
terms. We also present a different derivation, which better highlights that
such representation formulas are supported entirely on past null cones. This
generalization is a key component for extending Klainerman and Rodnianski's
breakdown criterion result for Einstein-vacuum spacetimes to Einstein-Maxwell
and Einstein-Yang-Mills spacetimes.Comment: 29 page
On Breakdown Criteria for Nonvacuum Einstein Equations
The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski
stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can
be further extended in time if the second fundamental form and the derivative
of the lapse of the foliation are uniformly bounded. This theorem and its proof
were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the
author's Ph.D. thesis. In this paper, we state the main results of the thesis,
and we summarize and discuss their proofs. In particular, we will discuss the
various issues resulting from nontrivial Ricci curvature and the coupling
between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6
(geometry of null cones
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