A Viscosity Method for the Min-Max Construction of Closed Geodesics

TL;DR

This paper introduces a viscosity method for constructing closed geodesics via min-max techniques on compact Riemannian manifolds, addressing regularity issues and analyzing index behavior in convergence.

Contribution

It develops a novel viscosity approach for min-max construction of closed geodesics and provides counter-examples to regularity in low dimensions.

Findings

Counter-examples to ε-regularity in dimensions 1 and 2

Proof of lower semi-continuity of the index during convergence

Development of a viscosity method for geodesic construction

Abstract

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension $1$ and $2$ to the $\varepsilon$-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index of our sequence of critical points converging towards a closed non-trivial geodesic.