An inequality for length and volume in the complex projective plane

TL;DR

This paper establishes a novel inequality connecting the length and volume of closed geodesics on area-minimizing surfaces in the complex projective plane, leveraging recent geometric regularity results and topological proofs.

Contribution

It introduces a new inequality relating length and volume in the complex projective plane, utilizing recent advances in geometric regularity and topological methods.

Findings

Proves a new inequality linking length and volume of geodesics.

Utilizes recent regularity results for area minimizers.

Employs the Kronheimer--Mrowka proof of the Thom conjecture.

Abstract

We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.