Bartnik mass via vacuum extensions

We construct asymptotically flat, scalar flat extensions of Bartnik data (ς,γ,H), where γ is a metric of positive Gauss curvature on a two-sphere ς, and H is a function that is either positive or identically zero on ς, such that the mass of the extension can be made arbitrarily close to the half area radius of (ς,γ). In the case of H 0, the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], 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 (ς,γ), for any metric γ 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 [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79-125] and a refined Shi-Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271-284].