82 research outputs found
Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity
We extend the idea and techniques in \cite{Miao} to study variational effect
of the boundary geometry on the ADM mass of an asymptotically flat manifold. We
show that, for a Lipschitz asymptotically flat metric extension of a bounded
Riemannian domain with quasi-convex boundary, if the boundary mean curvature of
the extension is dominated by but not identically equal to the one determined
by the given domain, we can decrease its ADM mass while raising its boundary
mean curvature. Thus our analysis implies that, for a domain with quasi-convex
boundary, the geometric boundary condition holds in Bartnik's minimal mass
extension conjecture \cite{Bartnik_energy}.Comment: 13 pages. This paper is closely related to math-ph/021202
Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface
We study a class of non-smooth asymptotically flat manifolds on which metrics
fails to be across a hypersurface . We first give an
approximation scheme to mollify the metric, then we prove that the Positive
Mass Theorem still holds on these manifolds if a geometric boundary condition
is satisfied by metrics separated by .Comment: 17 page
Total mean curvature, scalar curvature, and a variational analog of Brown-York mass
We study the supremum of the total mean curvature on the boundary of compact,
mean-convex 3-manifolds with nonnegative scalar curvature, and a prescribed
boundary metric. We establish an additivity property for this supremum and
exhibit rigidity for maximizers assuming the supremum is attained. When the
boundary consists of 2-spheres, we demonstrate that the finiteness of the
supremum follows from the previous work of Shi-Tam and Wang-Yau on the
quasi-local mass problem in general relativity. In turn, we define a
variational analog of Brown-York quasi-local mass without assuming that the
boundary 2-sphere has positive Gauss curvature.Comment: Final version; incorporates changes made for journa
Some functionals on compact manifolds with boundary
We analyze critical points of two functionals of Riemannian metrics on
compact manifolds with boundary. These functionals are motivated by formulae of
the mass functionals of asymptotically flat and asymptotically hyperbolic
manifolds
Nonexistence of NNSC fill-ins with large mean curvature
In this note we show that a closed Riemannian manifold does not admit a
fill-in with nonnegative scalar curvature if the mean curvature is point-wise
large. Similar result also holds for fill-ins with a negative scalar curvature
lower bound.Comment: 5 page
Scalar curvature rigidity with a volume constraint
Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's
conjecture, we prove a volume constrained scalar curvature rigidity theorem
which applies to the hemisphere.Comment: Theorem 1.7 and Theorem 5.1 strengthene
Static potentials on asymptotically flat manifolds
We consider the question whether a static potential on an asymptotically flat
3-manifold can have nonempty zero set which extends to the infinity. We prove
that this does not occur if the metric is asymptotically Schwarzschild with
nonzero mass. If the asymptotic assumption is relaxed to the usual assumption
under which the total mass is defined, we prove that the static potential is
unique up to scaling unless the manifold is flat. We also provide some
discussion concerning the rigidity of complete asymptotically flat 3-manifolds
without boundary that admit a static potential.Comment: introduction revised; an outline of a space-time approach adde
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