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Motivated by problems related to quasi-local mass in general relativity, westudy the static metric extension conjecture proposed by R. Bartnik\cite{Bartnik_energy}. We show that, for any metric on $\bar{B}_1$ that isclose enough to the Euclidean metric and has reflection invariant boundarydata, 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'sgeometric boundary condition \cite{Bartnik_energy} on $\partial B_1$.Comment: 20 page

On Existence of Static Metric Extensions in General Relativity