Lectures on Marked Length Spectrum Rigidity (preliminary Version)
نویسنده
چکیده
Let M be a closed Riemannian manifold whose sectional curvatures are all negative, and denote by C the set of free homotopy classes of closed curves in M . Negative curvature implies that in each free homotopy class, there is a unique closed geodesic. This defines a marked length spectrum function ` : C → R>0 which assigns to the class g the length `(g) of this closed geodesic. Burns and Katok asked whether the function ` determines M , up to isometry [7]. This question remains open in general, but has been solved completely for surfaces by Otal [27] and independently slightly later, but in greater generality by Croke [8]. In these notes, I’ll explain in several steps a proof of this marked length spectrum rigidity for negatively curved surfaces: Theorem 0.1 (Otal). Let S and S′ be closed, negatively curved surfaces with the same marked length spectrum. Then S is isometric to S′.
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تاریخ انتشار 2012