The marked length spectrum of a metric on a compact Riemannian manifold records the length of the shortest closed curve in each free homotopy class.
It is known that a negatively curved Riemannian metric on a compact surface is uniquely determined by its marked length spectrum up to isometry. My results show that under certain conditions on the excluded homotopy classes, a partial marked length spectrum also uniquely determines such a metric.
It is also known that an inequality between the marked length spectra of two negatively curved Riemannian metrics on a compact surface implies a corresponding inequality between the area with respect to the metrics. I will show that an inequality between the partial marked length spectra is enough to imply the same conclusion., In Copyright – Non-Commercial Use Permitted (InC-NC)