On minimal higher genus fillings

TL;DR

This paper establishes a lower bound on the area of a higher genus surface with boundary, relative to a disk, based on geodesic distance conditions, contributing to the understanding of minimal fillings in geometric topology.

Contribution

It proves a new inequality relating the area of a genus G surface to a disk under specific geodesic distance constraints, extending minimal filling results to higher genus surfaces.

Findings

Area inequality involving genus G and boundary conditions

Extension of minimal filling concepts to higher genus surfaces

Geodesic distance conditions imply area bounds

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 $$d_{(D,g_0)}(x,y) \geq d_{(M,g)}(x,y)$$ for all $x,y \in \partial M = \partial D$, then $$(1 + \frac{2 G}{\pi}) \textrm{Area}(M,g) \geq \textrm{Area}(D,g_0).$$