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
