Area minimizers and boundary rigidity of almost hyperbolic metrics

TL;DR

This paper extends previous work on boundary rigidity and minimal fillings, demonstrating that regions near hyperbolic metrics are uniquely determined by boundary distances and are minimal fillings, with a more invariant approach.

Contribution

It proves boundary rigidity and minimal filling properties for regions close to hyperbolic metrics, enhancing the understanding of metric rigidity near hyperbolic geometry.

Findings

Regions close to hyperbolic metrics are boundary distance rigid.

Such regions are strict minimal fillings.

The approach is made more invariant.

Abstract

This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.