Morse homology for the heat flow

TL;DR

This paper develops a Morse homology theory for the heat flow on the loop space of a closed Riemannian manifold, constructing an algebraic chain complex based on perturbed closed geodesics and heat flow trajectories.

Contribution

It introduces a novel Morse homology framework for heat flow on loop spaces, extending Floer theory concepts to this setting.

Findings

Constructed a chain complex generated by perturbed closed geodesics.

Defined a boundary operator counting heat flow trajectories between geodesics.

Established the algebraic structure linking heat flow dynamics to Morse homology.

Abstract

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.