Filling minimality of Finslerian 2-discs

TL;DR

This paper proves that Riemannian and Finsler metrics on the 2-disc with all geodesics minimal are minimal fillings of their boundary, extending previous results and generalizing Pu's isosystolic inequality to Finsler metrics.

Contribution

It removes the convex boundary assumption and extends minimal filling results to Finsler metrics with Holmes-Thompson area, broadening the scope of geometric inequalities.

Findings

Every geodesically minimal Riemannian 2-disc is a minimal filling.

Generalization of Pu's inequality to Finsler metrics.

Extension to Finsler metrics with Holmes-Thompson volume.

Abstract

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for Holmes-Thompson and Busemann definitions of Finsler area.