On minimal higher genus fillings

Abstract

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that interior points are connected by unique geodesics and $$d_{(D,g_0)}(x,y)\ge d_{(M,g)}(x,y)$$ for all $x,y\in \partial M=\partial D$, then $$(1+\frac{2G}{\pi })\text{Area}(M,g)\ge \text{Area}(D,g_0).$$

16 pages, 3 figures