Abstract
Commun. Math. Phys. 241(1) , 27-46 (2003) 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 $\partial B_1$.
