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

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 $g^{\prime }$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g^{\prime })\ge vol(D^n,g)$. 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.

arXiv comment13 pages, v3: minor corrections and clarifications, to appear in Geometriae Dedicata

References in corpus (1)

• Filling minimality of Finslerian 2-discs

Cited by in corpus (4)

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