A counterexample to marked length spectrum semi-rigidity

TL;DR

This paper constructs a specific example of a negatively curved surface where all closed geodesics are lengthened by a perturbation, yet no diffeomorphism can uniformly contract tangent vectors, challenging assumptions in spectral geometry.

Contribution

It provides a counterexample to the semi-rigidity of the marked length spectrum for negatively curved surfaces.

Findings

Existence of a perturbation increasing all closed geodesic lengths

No diffeomorphism can contract all tangent vectors after perturbation

Challenges assumptions in length spectrum rigidity

Abstract

Given a closed orientable negatively curved Riemannian surface $(M,g)$, we show how to construct a perturbation $(M,g^\prime)$ such that each closed geodesic becomes longer, and yet there is no diffeomorphism $f : (M,g^\prime) \rightarrow (M,g)$ which contracts every tangent vector.