Marked Poincar{é} rigidity near hyperbolic metrics

Abstract

Let $M$ be a compact manifold without boundary equipped with a Riemannian metric $g$ of negative curvature. In this paper, we introduce the marked Poincar{é} determinant (MPD), a homothety invariant of $g$ depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in $M$ a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local rigidity result: if $g$ is sufficiently close to a hyperbolic metric $g_0$ and both metrics have the same MPD and the same normalized total scalar curvature, then they are homothetic. We also prove that $g_0$ is locally determined by the MPD alone when its Lichnerowicz Laplacian is injective on trace-free divergence-free symmetric 2-tensors.

Substantial revision due to an error before (5.4) on p.16 of v1, affecting Theorems 1.6, 1.8, and hence 1.3. Theorem 1.3 v2 now assumes equality of total scalar curvature in addition to equality of MPD, but holds in all dimensions. Theorem 1.3 v1 is open; Section 6 v2 discusses examples where it may hold. Theorem 1.8 v1 is removed; its statement is correct but not needed for Theorem 1.3