Local unmarked length spectrum rigidity for hyperbolic surfaces

Abstract

Let $(M,g)$ be a closed negatively curved surface. If $g$ is strictly $\frac{1}{9}$-pinched and has the same unmarked length spectrum as a hyperbolic metric, we show that $g$ is hyperbolic. As a consequence, we show that any hyperbolic metric on a surface admits a $C^2$-neighborhood in the space of metrics in which it is characterized by its unmarked length spectrum, up to isometry.

arXiv commentcomments welcome, added Corollary 1.5 on commensurable hyperbolic metrics in v2