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

TL;DR

This paper explores the geometric and topological properties of surfaces with boundary under $rak{g}$-stability conditions, providing new insights into scalar curvature, area estimates, and boundary behaviors, including capillary cases.

Contribution

It introduces new results linking $rak{g}$-stability with positive scalar curvature metrics, area bounds, and topology classification for surfaces with boundary, extending to capillary boundaries.

Findings

Existence of positive scalar curvature metrics with minimal boundary for $rak{g}$-stable hypersurfaces.

Derived area estimates and topological classifications for $rak{g}$-stable surfaces with free boundary.

Extended free boundary results to capillary boundary conditions.

Abstract

In this paper, we investigate the geometric properties associated with the $\mathfrak{g}$-stability of surfaces with boundary whose null expansion satisfies $\Theta^{+} = h \geq 0$. 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.