Relative isoperimetric inequality on a curved surface

Abstract

Let $C$ be a closed convex set on a complete simply connected surface $S$ whose Gaussian curvature is bounded above by a nonpositive constant $K$. For a relatively compact subset $\Omega \subset S \sim C$, we obtain the sharp relative isoperimeric inequality $2\pi \mathrm{Area}(\Omega )-K\mathrm{Area}(\Omega )^{2} \leq \mathrm{Length}(\partial \Omega \sim \partial C)^{2}$. And we also have a similar partial result with positive Gaussian curvature bound.