Marked length spectrum rigidity in nonpositive curvature with singularities

TL;DR

This paper proves that for certain nonpositively curved surfaces with cone singularities, the marked length spectrum uniquely determines the metric, extending previous rigidity results to include singularities and weaker conditions.

Contribution

It establishes marked length spectrum rigidity for nonpositively curved surfaces with cone singularities, broadening the scope of prior rigidity theorems to include singular metrics.

Findings

Marked length spectrum determines the metric in the specified class

Rigidity holds even with cone angles greater than 2π

Weaker conditions like no conjugate points are sufficient under additional assumptions

Abstract

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2\pi$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'