A local systolic inequality and Gromov's filling area conjecture

TL;DR

This paper investigates Gromov's filling area conjecture, aiming to establish volume bounds and a local systolic inequality, but contains a computational error in its proof of the latter.

Contribution

It attempts to prove a local systolic inequality and volume bounds related to Gromov's conjecture, highlighting the challenges in such geometric inequalities.

Findings

Proposed that fillings with volume less than 2π are unattainable

Attempted to establish a lower volume estimate via a local systolic inequality

Identified a computational mistake in the proof of the systolic inequality

Abstract

The article treats some questions around Gromov's filling area conjecture. It intended to show that any filling with volume $< 2 \pi$ would not be attained and to show a local systolic inequality, implying an a priori lower estimate on the total volume. The proof of both assertions, the second of which is wrong, suffered from a computational mistake.