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

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

2011-09-19

arxiv.org/abs/1109.4091v2

Mathematics

Differential Geometry

12 pages. Changes in version 2: more detailed proofs and an appendix with a technical lemma

Scientific paper

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.

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