Bartnik mass via vacuum extensions

TL;DR

This paper constructs vacuum extensions of Bartnik data with scalar flatness and arbitrarily close mass to the half area radius, advancing understanding of quasi-local mass and the Riemannian Penrose inequality in general relativity.

Contribution

It introduces a method to produce asymptotically flat, scalar flat extensions with near-optimal mass, extending previous results to vacuum initial data with vanishing scalar curvature.

Findings

Mass can be made arbitrarily close to half the area radius.

Constructs vacuum initial data with apparent horizons matching positive Gauss curvature metrics.

Extends Mantoulidis and Schoen's theorem to scalar flat, vacuum extensions.

Abstract

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}.