On Existence of Static Metric Extensions in General Relativity

Abstract

Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik \cite{Bartnik_energy}. We show that, for any metric on ${\bar{B}}_1$ that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat {\em static} metric extension in $M={\mathbb{R}}^3\setminus B_1$ such that it satisfies Bartnik's geometric boundary condition \cite{Bartnik_energy} on $\partial B_1$.

20 pages

Cited by in corpus (10)

• On the volume functional of compact manifolds with boundary with constant scalar curvature
• On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature
• A Remark on Boundary Effects in Static Vacuum Initial Data sets
• On the Bartnik extension problem for the static vacuum Einstein equations
• The Bartnik-Bray outer mass of small metric spheres in time-symmetric 3-slices
• New asymptotically flat static vacuum metrics with near Euclidean boundary data
• Three Quasi-Local Masses
• Mean-stable surfaces in Static Einstein-Maxwell theory
• Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric
• Local Well-posedenss of the Bartnik Static Extension Problem near Schwarzschild spheres