Curve shortening and the topology of closed geodesics on surfaces

TL;DR

This paper introduces a Conley index framework for analyzing the topology of closed geodesics on surfaces, establishing existence results for geodesics with specific knot types via curve shortening flow.

Contribution

It develops a novel Conley index approach for the curve shortening flow to prove the existence of closed geodesics with prescribed flat knot types on surfaces.

Findings

Existence of closed geodesics with given flat knot types

Nontrivial Conley index implies geodesic existence

Framework applies to any Riemannian metric on surfaces

Abstract

We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.