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Some results on the $\mathfrak{g}$-stability of surfaces with boundary

Sanghun Lee

TL;DR

This work extends the analysis of generalized marginally outer trapped surfaces to surfaces with boundary by introducing and exploiting the notion of $\mathfrak{g}$-stability for $\Theta^{+}=h\ge0$. It proves that a $\mathfrak{g}$-stable free-boundary hypersurface under the dominant boundary energy condition and appropriate energy bounds carries a metric of positive scalar curvature with minimal boundary, and in dimension three derives an area bound and disk topology with rigidity when equality is achieved. To handle capillary boundaries, the paper introduces the tilted dominant boundary energy condition (TDBEC) and shows that the same PSC-minimal boundary conclusions and rigidity results hold, with the capillary angle $\theta$ entering the boundary conditions. The results combine eigenvalue analysis of a generalized stability operator, Robin boundary problems, and Gauss-Bonnet-type arguments to connect spacetime energy conditions to intrinsic-extrinsic geometry of boundary-bearing surfaces, thereby extending MOTS rigidity phenomena to broader boundary configurations.

Abstract

In this paper, we investigate the geometric properties associated with the $\mathfrak{g}$-stability of surfaces with boundary whose null expansion satisfies $Θ^{+} = h \geq 0$. First, we show that a $\mathfrak{g}$-stable hypersurface with free boundary admits a metric of positive scalar curvature with minimal boundary under suitable conditions. Second, for $\mathfrak{g}$-stable surfaces with free boundary, we derive an area estimate and determine the topology of the surface. Finally, we extend our free boundary results to the case of capillary boundary.

Some results on the $\mathfrak{g}$-stability of surfaces with boundary

TL;DR

This work extends the analysis of generalized marginally outer trapped surfaces to surfaces with boundary by introducing and exploiting the notion of -stability for . It proves that a -stable free-boundary hypersurface under the dominant boundary energy condition and appropriate energy bounds carries a metric of positive scalar curvature with minimal boundary, and in dimension three derives an area bound and disk topology with rigidity when equality is achieved. To handle capillary boundaries, the paper introduces the tilted dominant boundary energy condition (TDBEC) and shows that the same PSC-minimal boundary conclusions and rigidity results hold, with the capillary angle entering the boundary conditions. The results combine eigenvalue analysis of a generalized stability operator, Robin boundary problems, and Gauss-Bonnet-type arguments to connect spacetime energy conditions to intrinsic-extrinsic geometry of boundary-bearing surfaces, thereby extending MOTS rigidity phenomena to broader boundary configurations.

Abstract

In this paper, we investigate the geometric properties associated with the -stability of surfaces with boundary whose null expansion satisfies . First, we show that a -stable hypersurface with free boundary admits a metric of positive scalar curvature with minimal boundary under suitable conditions. Second, for -stable surfaces with free boundary, we derive an area estimate and determine the topology of the surface. Finally, we extend our free boundary results to the case of capillary boundary.
Paper Structure (4 sections, 7 theorems, 43 equations)

This paper contains 4 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results for free boundary
  4. Results for capillary boundary

Key Result

Theorem 1.1

Let $(M^{n+1}, g, A^{M})$ be an $(n+1)$-dimensional initial data set with boundary, and let $\Sigma^{n}$ be a compact $n$-dimensional $\mathfrak{g}$-stable hypersurface with free boundary such that the null expansion satisfies $\Theta^{+} = h \in C^{\infty}(\Sigma)$, with $h \geq 0$. Suppose that $( then $\Sigma$ admits a metric of positive scalar curvature with minimal boundary.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Definition 2.1 in GM1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 3.1: Lemma 3.1 and Lemma 3.2 in M2
  • Lemma 4.1: Lemma 3.1 in LEE and Lemma 3.2 in M2
  • ...and 1 more