Problems of unique determination of domains by the relative metrics on their boundaries

TL;DR

This survey explores the challenges and conditions for uniquely determining nonconvex domain boundaries using relative boundary metrics, including rigidity and local isometry conditions, with references to recent research results.

Contribution

It synthesizes recent results on boundary determination problems, highlighting rigidity conditions and local isometry criteria for nonconvex domains and their boundaries.

Findings

Results on unique determination of domains by boundary metrics

Rigidity conditions for boundaries in Riemannian manifolds

Criteria for local isometry of boundaries

Abstract

This survey is devoted to discussing the problems of the unique determination of surfaces that are the boundaries of (generally speaking) nonconvex domains. First (in Sec. 2) we examine some results on the problem of the unique determination of domains by the relative metrics of the boundaries. Then, in Sec. 3, we study rigidity conditions for the boundaries of submanifolds in a Riemannian manifold. The final part (Sec. 4) is concerned with the unique determination of domains by the condition of the local isometry of boundaries in the relative metrics. The survey in particular contains the results of arXiv:1401.7295, arXiv:1305.6169, arXiv:1511.04235.