A Brief Summary of Otal’s Proof of Marked Length Spectrum Rigidity

We outline Otal’s proof of marked length spectrum rigidity for negatively curved surfaces. We omit all technical details, and refer the interested reader to the original [Ota90] or the course notes [Wil] for details, and to [Cro90] for different approach. (Actually the course notes [Wil] combine the approaches in [Ota90,Cro90].) The author thanks Amie Wilkinson for explaining this proof to him. This informal note was written while the author was Amie Wilkinson’s teaching assistant for her course on the same topic at the Park City Math Institute, 2012. The author thanks Jenny Wilson for producing the figures. Consider two negatively curved closed surfaces S and S ′. Fix a homeomorphism from S to S ′, or alternatively consider S and S ′ to be two Riemannian structures on the same topological surface. Due to negative curvature, every closed curve is homotopic to a unique closed geodesic, called the geodesic representative of the homotopy class. Let C denote the set of homotopy classes of closed curves. The marked length spectrum of S is defined as the function `S : C → R>0 which assigns to each homotopy class of curve the length of its geodesic representative. Theorem 1 (Otal, Annals 1990). Let S and S ′ be two negatively curved closed marked surfaces. If S and S ′ have the same marked length spectrum, they are isometric. Step 1: Coarse geometry gives a correspondence of geodesics. Let S̃ and S̃ ′ denote the universal covers of S and S ′. The homeomorphism Id : S → S ′ lifts to a homeomorphism Ĩd : S̃ → S̃ ′, which is in fact a quasi-isometry. Again due to negative curvature, both S̃ and S̃ ′ have boundaries, which are homeomorphic to a circle. The 1