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

Abstract

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

11p