On the Hofer geometry injectivity radius conjecture

TL;DR

This paper investigates the injectivity radius conjecture in Hofer geometry for Hamiltonian groups of surfaces, proving contractibility of certain loops and absence of specific geodesics, with implications for the group's curvature properties.

Contribution

It verifies variants of the injectivity radius conjecture for $Ham(S^2)$ and $Ham( ext{surface})$, showing contractibility of loops below a certain length and ruling out specific Morse index geodesics.

Findings

Loops with length less than half the area are contractible in the Hofer metric.

No positive Morse index geodesics exist below a certain length in these groups.

Results suggest connections between geodesic properties and curvature of Hamiltonian diffeomorphism groups.

Abstract

We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev \cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of $Ham (S^2)$ and $Ham(\Sigma, \omega)$, for $\Sigma$ a closed positive genus surface. In particular we show that any loop in $Ham (S^2)$, respectively $Ham(\Sigma, \omega)$ with $L ^{+}$ Hofer length less than $area(S ^{2} )/2$, respectively any $L ^{+} $ length is contractible through ($L ^{+} $) Hofer shorter loops, in the $C ^{\infty} $ topology. We also prove some stronger variants of this statement on the loop space level. One dynamical type corollary is that there are no smooth, positive Morse index (Ustilovsky) geodesics, in $Ham (S^2)$, respectively in $Ham(\Sigma, \omega)$ with $L ^{+} $ Hofer length less than $area (S ^{2} )/2$, respectively any length. The above condition on the geodesics can be expanded as an explicit and elementary dynamical condition on the associated Hamiltonian flow. We also give some speculations on connections of this later result with curvature properties of the Hamiltonian diffeomorphism group of surfaces.