Local monotonicity of Riemannian and Finsler volume with respect to boundary distances

TL;DR

This paper proves that the volume of simple Riemannian and Finsler metrics on a disk is locally monotone with respect to boundary distances, and uses this to establish injectivity of the geodesic ray transform for Finsler metrics.

Contribution

It establishes local monotonicity of volume with respect to boundary distances for Riemannian and Finsler metrics, providing a new proof of geodesic ray transform injectivity.

Findings

Volume increases with boundary distance under perturbations

Monotonicity holds for both Riemannian and Finsler metrics

New proof of geodesic ray transform injectivity

Abstract

We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermore, the same holds for Finsler metrics and the Holmes--Thompson definition of volume. As an application, we give a new proof of the injectivity of the geodesic ray transform for a simple Finsler metric.