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

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