Horizons in noncompact fill-ins of nonnegative scalar curvature

TL;DR

This work studies horizon formation in noncompact, nonnegatively curved fill-ins of $N=\Sigma\times(-\infty,0]$ with $\Sigma\cong S^2$, connecting to Bartnik mass via a mu-bubble/mass-systole framework. It constructs admissible extensions, applies Zhu's mass-systole inequality to obtain a lower bound $\mathfrak{m}(\tilde{g})\ge\sqrt{A(\tilde{g})/(16\pi)}$ and shows $A(\tilde{g})=A(g)$ when appropriate, implying either a horizon exists or $A_{-\infty}(g)=A(g)$. The paper then derives concrete horizon criteria in terms of $A(g)$, $A_{-\infty}(g)$, and the boundary data $(\Sigma,\sigma,H)$, including roundsphere and positive Gauss curvature cases, and capacity/gradient conditions that force horizons. These results advance the understanding of quasi-local mass, noncompact fill-ins, and Penrose-type inequalities in Riemannian geometry.

Abstract

Given a complete Riemannian metric of nonnegative scalar curvature on $Σ\times (-\infty, 0 ] $, where $Σ$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the boundary.

Theorems & Definitions (15)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Proposition 1.1
• Proposition 1.2
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.4