Volume comparison via boundary distances

TL;DR

This paper explores the relationship between Gromov's filling volumes and boundary rigidity in Riemannian geometry, using minimal surface representations to establish metric determination from boundary distances and discussing recent advances and related Finsler geometry problems.

Contribution

It introduces a novel approach linking filling volumes and boundary rigidity through minimal surface representations, advancing understanding of metric determination from boundary data.

Findings

Established connections between filling volumes and boundary rigidity

Demonstrated the effectiveness of minimal surface representations in rigidity proofs

Discussed recent results and open problems in Finsler geometry

Abstract

The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to represent Riemannian metrics by minimal surfaces in a Banach space and to prove rigidity by studying the equality case in a filling volume inequality. I discuss recent results obtained with this approach and related problems in Finsler geometry.