926 research outputs found
Invariants of the harmonic conformal class of an asymptotically flat manifold
Consider an asymptotically flat Riemannian manifold of dimension with nonempty compact boundary. We recall the harmonic conformal class
of the metric, which consists of all conformal rescalings given by a
harmonic function raised to an appropriate power. The geometric significance is
that every metric in has the same pointwise sign of scalar curvature.
For this reason, the harmonic conformal class appears in the study of general
relativity, where scalar curvature is related to energy density. Our purpose is
to introduce and study invariants of the harmonic conformal class. These
invariants are closely related to constrained geometric optimization problems
involving hypersurface area-minimizers and the ADM mass. In the final section,
we discuss possible applications of the invariants and their relationship with
zero area singularities and the positive mass theorem.Comment: 26 pages, 2 figure
On the lower semicontinuity of the ADM mass
The ADM mass, viewed as a functional on the space of asymptotically flat
Riemannian metrics of nonnegative scalar curvature, fails to be continuous for
many natural topologies. In this paper we prove that lower semicontinuity holds
in natural settings: first, for pointed Cheeger--Gromov convergence (without
any symmetry assumptions) for , and second, assuming rotational symmetry,
for weak convergence of the associated canonical embeddings into Euclidean
space, for . We also apply recent results of LeFloch and Sormani to
deal with the rotationally symmetric case, with respect to a pointed type of
intrinsic flat convergence. We provide several examples, one of which
demonstrates that the positive mass theorem is implied by a statement of the
lower semicontinuity of the ADM mass.Comment: 18 pages, 4 figure
Smoothing the Bartnik boundary conditions and other results on Bartnik's quasi-local mass
Quite a number of distinct versions of Bartnik's definition of quasi-local
mass appear in the literature, and it is not a priori clear that any of them
produce the same value in general. In this paper we make progress on
reconciling these definitions. The source of discrepancies is two-fold: the
choice of boundary conditions (of which there are three variants) and the
non-degeneracy or "no-horizon" condition (at least six variants). To address
the boundary conditions, we show that given a 3-dimensional region of
nonnegative scalar curvature () extended in a Lipschitz fashion
across to an asymptotically flat 3-manifold with
(also holding distributionally along ), there exists a
smoothing, arbitrarily small in norm, such that and the
geometry of are preserved, and the ADM mass changes only by a small
amount. With this we are able to show that the three boundary conditions yield
equivalent Bartnik masses for two reasonable non-degeneracy conditions. We also
discuss subtleties pertaining to the various non-degeneracy conditions and
produce a nontrivial inequality between a no-horizon version of the Bartnik
mass and Bray's replacement of this with the outward-minimizing condition.Comment: 25 pages, 3 figure
Lower semicontinuity of the ADM mass in dimensions two through seven
The semicontinuity phenomenon of the ADM mass under pointed (i.e., local)
convergence of asymptotically flat metrics is of interest because of its
connections to nonnegative scalar curvature, the positive mass theorem, and
Bartnik's mass-minimization problem in general relativity. In this paper, we
extend a previously known semicontinuity result in dimension three for
pointed convergence to higher dimensions, up through seven, using recent work
of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose
inequality of H. Bray and D. Lee). For a technical reason, we restrict to the
case in which the limit space is asymptotically Schwarzschild. In a separate
result, we show that semicontinuity holds under weighted, rather than pointed,
convergence, in all dimensions , with a simpler proof
independent of the positive mass theorem. Finally, we also address the
two-dimensional case for pointed convergence, in which the asymptotic cone
angle assumes the role of the ADM mass.Comment: 23 pages. Comments welcome
Time flat surfaces and the monotonicity of the spacetime Hawking mass
We identify a condition on spacelike 2-surfaces in a spacetime that is
relevant to understanding the concept of mass in general relativity. We prove a
formula for the variation of the spacetime Hawking mass under a uniformly area
expanding flow and show that it is nonnegative for these so-called "time flat
surfaces." Such flows generalize inverse mean curvature flow, which was used by
Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black
hole. A flow of time flat surfaces may have connections to the problem in
general relativity of bounding the mass of a spacetime from below by the
quasi-local mass of a spacelike 2-surface contained therein.Comment: 23 pages; sign error fixed from previous version, statement of
Theorem 1.1 changed accordingl
A geometric theory of zero area singularities in general relativity
The Schwarzschild spacetime metric of negative mass is well-known to contain
a naked singularity. In a spacelike slice, this singularity of the metric is
characterized by the property that nearby surfaces have arbitrarily small area.
We develop a theory of such "zero area singularities" in Riemannian manifolds,
generalizing far beyond the Schwarzschild case (for example, allowing the
singularities to have nontrivial topology). We also define the mass of such
singularities. The main result of this paper is a lower bound on the ADM mass
of an asymptotically flat manifold of nonnegative scalar curvature in terms of
the masses of its singularities, assuming a certain conjecture in conformal
geometry. The proof relies on the Riemannian Penrose Inequality. Equality is
attained in the inequality by the Schwarzschild metric of negative mass. An
immediate corollary is a version of the Positive Mass Theorem that allows for
certain types of incomplete metrics.Comment: 36 pages, 9 figure
On curves with nonnegative torsion
We provide new results and new proofs of results about the torsion of curves
in . Let be a smooth curve in that is the
graph over a simple closed curve in with positive curvature. We
give a new proof that if has nonnegative (or nonpositive) torsion,
then has zero torsion and hence lies in a plane. Additionally, we
prove the new result that a simple closed plane curve, without any assumption
on its curvature, cannot be perturbed to a closed space curve of constant
nonzero torsion. We also prove similar statements for curves in Lorentzian
which are related to important open questions about time
flat surfaces in spacetimes and mass in general relativity.Comment: 13 pages, 7 figures; references added in second versio
Lower semicontinuity of mass under convergence and Huisken's isoperimetric mass
Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar
curvature with outermost minimal boundary, converging in the pointed
Cheeger--Gromov sense to an asymptotically flat limit space, we show that the
total mass of the limit is bounded above by the liminf of the total masses of
the sequence. In other words, total mass is lower semicontinuous under such
convergence. In order to prove this, we use Huisken's isoperimetric mass
concept, together with a modified weak mean curvature flow argument. We include
a brief discussion of Huisken's work before explaining our extension of that
work. The results are all specific to three dimensions.Comment: 30 pages, 5 figure
Time flat surfaces and the monotonicity of the spacetime Hawking mass II
In this sequel paper we give a shorter, second proof of the monotonicity of
the Hawking mass for time flat surfaces under spacelike uniformly area
expanding flows in spacetimes that satisfy the dominant energy condition. We
also include a third proof which builds on a known formula and describe a class
of sufficient conditions of divergence type for the monotonicity of the Hawking
mass. These flows of surfaces may have connections to the problem in general
relativity of bounding the total mass of a spacetime from below by the
quasi-local mass of spacelike 2-surfaces in the spacetime.Comment: 19 pages, 1 figur
Unfolding a degeneracy point of two unbound states: Crossings and anticrossings of energies and widths
We show that when an isolated doublet of unbound states of a physical system
becomes degenerate for some values of the control parameters of the system, the
energy hypersurfaces representing the complex resonance energy eigenvalues as
functions of the control parameters have an algebraic branch point of rank one
in parameter space. Associated with this singularity in parameter space, the
scattering matrix, S_l(E), and the Green's function, G_l^(+)(k; r,r'), have one
double pole in the unphysical sheet of the complex energy plane. We
characterize the universal unfolding or deformation of a typical degeneracy
point of two unbound states in parameter space by means of a universal
2-parameter family of functions which is contact equivalent to the pole
position function of the isolated doublet of resonances at the exceptional
point and includes all small perturbations of the degeneracy condition up to
contact equivalence.Comment: 6 pages and 3 eps figure
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