Exterior Schwarzschild initial data for degenerate apparent horizons

Abstract

In this note we show that if $g$ is a smooth Riemannian metric on ${\mathbb{S}}^2$ such that the first eigenvalue of the operator $L_g:=-{\Delta }_g+K_g$ satisfies ${\lambda }_1(L_g)=0$ then $({\mathbb{S}}^2,g)$ arises as an apparent horizon in an asymptotically flat initial data set with ADM mass arbitrarily close to the associated Hawking mass $\sqrt{\text{area}({\mathbb{S}}^2,g)/16\pi }$. In particular, this determines the Bartnik quasilocal mass (introduced by Bartnik \cite{Bartnik} in 1989) associated with $({\mathbb{S}}^2,g)$ in this setting. We prove these by modifying the construction of Mantoulidis-Schoen \cite{MS} who proved the same results in the case ${\lambda }_1(L_g)>0$. It follows that ${\lambda }_1(g)\ge 0$ is necessary and sufficient for $({\mathbb{S}}^2,g)$ to arise from an apparent horizon in an asyptotically flat space-time under the dominant energy condition and in the time symmetric setting, and that the Bartnik mass of the horizon is $\sqrt{\text{area}({\mathbb{S}}^2,g)/16\pi }$.

7 pages; details added to introduction and statements of Theorem 1.1 and Lemma 2.2