On Existence of Static Metric Extensions in General Relativity
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 $M = \R^3 \setminus B_1$ such that it satisfies Bartnik's geometric boundary condition \cite{Bartnik_energy} on $\partial B_1$.
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