Existence of Good Sweepouts on Closed Manifolds

TL;DR

This paper develops estimates for harmonic map heat flow from circles into closed manifolds to construct sweepouts where high-energy curves are close to closed geodesics, aiding in geometric analysis.

Contribution

It introduces a method to produce sweepouts with curves near geodesics, leveraging harmonic map heat flow estimates, advancing the understanding of geodesic existence.

Findings

Constructed sweepouts with curves close to geodesics

Established estimates for harmonic map heat flow

Connected high-energy curves to closed geodesics

Abstract

In this note we establish estimates for the harmonic map heat flow from $S^1$ into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.