Closed geodesics in Alexandrov spaces of curvature bounded from above

TL;DR

This paper establishes local energy convexity for maps into CAT(K) spaces, enabling the extension of geodesic existence results to Alexandrov spaces with curvature bounded above, generalizing classical theorems.

Contribution

It introduces a new energy convexity property and extends the width-sweepout method to prove the existence of closed geodesics in Alexandrov spaces.

Findings

Proves local energy convexity in CAT(K) spaces.

Extends geodesic existence theorems to Alexandrov spaces.

Generalizes Birkhoff-Lyusternik theorem.

Abstract

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.