  We construct asymptotically flat, scalar flat extensions of Bartnik data
$(\Sigma, \gamma, H)$, where $\gamma$ is a metric of positive Gauss curvature
on a two-sphere $\Sigma$, and $H$ is a function that is either positive or
identically zero on $\Sigma$, such that the mass of the extension can be made
arbitrarily close to the half area radius of $(\Sigma, \gamma)$.
  In the case of $H \equiv 0$, the result gives an analogue of a theorem of
Mantoulidis and Schoen, but with extensions that have vanishing scalar
curvature. In the context of initial data sets in general relativity, the
result produces asymptotically flat, time-symmetric, vacuum initial data with
an apparent horizon $(\Sigma, \gamma)$, for any metric $\gamma$ with positive
Gauss curvature, such that the mass of the initial data is arbitrarily close to
the optimal value in the Riemannian Penrose inequality.
  The method we use is the Shi-Tam type metric construction from
\cite{ShiTam02} and a refined Shi-Tam monotonicity, found by the first named
author in \cite{Miao09}.
