On the Bartnik conjecture for the static vacuum Einstein equations

TL;DR

This paper proves the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary metric and mean curvature, partially resolving Bartnik's conjecture in mathematical physics.

Contribution

It establishes the existence of solutions with prescribed boundary data for the static vacuum Einstein equations, advancing understanding of the Bartnik conjecture.

Findings

Existence of solutions for given boundary metric and mean curvature.

Dependence of the solution on boundary data.

Partial resolution of Bartnik's conjecture.

Abstract

We prove that given any smooth metric $\gamma$ and smooth positive function $H$ on $S^{2}$, there is a constant $\lambda > 0$, depending on $(\gamma, H)$, and an asymptotically flat solution $(M, g, u)$ of the static vacuum Einstein equations on $M = {\mathbb R}^{3} \setminus B^{3}$, such that the induced metric and mean curvature of $(M, g, u)$ at $\partial M$ are given by $(\gamma, \lambda H)$. This gives a partial resolution of a conjecture of Bartnik.