Stability of Anosov Hamiltonian Structures

TL;DR

This paper investigates the stability of Anosov Hamiltonian structures on energy levels of negatively curved manifolds with twisted symplectic forms, establishing conditions under which such structures are contact and thus stable.

Contribution

It proves that for odd dimensions, Anosov Hamiltonian structures are stable if and only if they are contact, extending to even dimensions under pinching conditions, and applies to negatively curved manifolds.

Findings

Hamiltonian structures are stable iff they are contact in odd dimensions.

In even dimensions, stability requires the flow to be 1/2-pinched.

Negatively curved, strictly 1/4-pinched manifolds with non-exact 2-forms have unstable Hamiltonian structures.

Abstract

Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $\Sigma_{k}:=H^{-1}(k)$ of $H$. Suppose $\Sigma_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $\Sigma_{k}$ is Anosov with $C^{1}$ weak bundles then the Hamiltonian structure $(\Sigma_{k}$ is stable if and only if it is contact. If $n$ is even and in addition the flow is assumed to be 1/2-pinched then the same conclusion holds. As a corollary we deduce that if $g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the twisted symplectic structure is not exact then the Hamiltonian structure $(\Sigma_{k}$ is never stable for all sufficiently large $k$.