Proof of the index conjecture in Hofer geometry

TL;DR

This paper proves a conjecture linking the Morse index of certain geodesics in Hofer geometry to the Conley-Zehnder index of associated periodic orbits, providing a simplified proof of a significant theoretical relationship.

Contribution

It offers a straightforward proof of a generalized index conjecture in Hofer geometry connecting Morse and Conley-Zehnder indices.

Findings

Established the relation between Morse index and Conley-Zehnder index for non-degenerate Ustilovsky geodesics.

Simplified the proof of the index conjecture in Hofer geometry.

Extended the conjecture to a broader class of geodesics.

Abstract

Let $\gamma$ be a non-degenerate Ustilovsky geodesic in $Ham (M, \omega)$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $ \gamma$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.