Let (M,g) be a closed negatively curved manifold. We introduce a new invariant, the Marked Poincaré determinant (MPD) which 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. This invariant can be seen as a weighted version (by a function called the unstable Jacobian) of a well-known invariant of (M,g): the marked length spectrum. We prove a local MPD rigidity result in dimension 3: if g is sufficiently close to a hyperbolic metric g0 and both metrics have the same MPD, then they are homothetic (i.e. isometric up to rescalling).The proof relies on a geometric fact of independent interest, namely, we show the Lichnerowicz Laplacian of g0 is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

This is joint work with Karen Butt, Alena Erchenko, Thibault Lefeuvre and Amie Wilkinson.