A Systolic Inequality for the Filling Area Conjecture

TL;DR

This paper establishes an upper bound on the systolic ratio for orientable isometric fillings of the circle, linking it to genus, and confirms the Filling Area Conjecture for large genus in this class.

Contribution

It provides a new genus-dependent upper bound on the systolic ratio and verifies the Filling Area Conjecture for large genus orientable isometric fillings.

Findings

Upper bound on systolic ratio depending on genus

Filling Area Conjecture holds for large genus cases

Application to fillings with a lower bound on systole

Abstract

We prove an upper bound on the systolic ratio of an orientable isometric filling of the circle equipped with a Riemannian metric. The bound depends only on the genus of isometric filling. We also apply the bound to the class of orientable isometric filling with a certain lower bound on the systole. We deduce that the Filling Area Conjecture holds true for this class when the genus is sufficiently large.