The marked length spectrum of a projective manifold or orbifold

TL;DR

This paper investigates the relationship between the marked length spectrum and the projective structure of convex real projective orbifolds, showing that the spectrum determines the structure up to duality and revealing non-uniqueness in isospectral cases.

Contribution

It proves that the marked Hilbert length spectrum determines the projective structure only up to projective duality, highlighting non-uniqueness in isospectral convex projective manifolds.

Findings

Marked length spectrum determines projective structure up to duality

Existence of non-isometric isospectral convex projective orbifolds

Connection between Hilbert and hyperbolic metrics in special cases

Abstract

A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to projective duality. A corollary is the existence of non-isometric diffeomorphic strictly convex projective manifolds (and orbifolds) that are isospectral. The corollary follows from work of Goldman and Choi, and Benoist.