STABILITY OF ANOSOV HAMILTONIAN STRUCTURES

Abstract

Let (Mn, g) denote a closed Riemannian manifold (n ≥ 3) which admits a metric of negative curvature (not necessarily equal to g). Let ω1 := ω0 + π*σ denote a twisted symplectic form on TM, where σ ∈ Ω2(M) is a closed 2-form and ω0 is the symplectic structure on TM obtained by pulling back the canonical symplectic form dx ∧ dp on T*M via the Riemannian metric. Let Σk be the hypersurface [Formula: see text]. We prove that if n is odd and the Hamiltonian structure (Σk, ω1) is Anosov with C1 weak bundles, then (Σk, ω1) is stable if and only if it is contact. If n is even and in addition the Hamiltonian structure is 1/2-pinched, then the same conclusion holds. As a corollary, we deduce that if g is negatively curved, strictly 1/4-pinched and σ is not exact then the Hamiltonian structure (Σk, ω1) is never stable for all sufficiently large k.