Boundary rigidity and filling volume minimality of metrics close to a flat one

Abstract

We say that a Riemannian manifold $(M, g)$ with a non-empty boundary $\partial M$ is a minimal orientable filling if, for every compact orientable $(\widetilde M,\tilde g)$ with $\partial \widetilde M=\partial M$, the inequality $ d_{\tilde g}(x,y) \ge d_g(x,y)$ for all $x,y\in\partial M$ implies $ \operatorname{vol}(\widetilde M,\tilde g) \ge \operatorname{vol}(M,g) .$ We show that if a metric $g$ on a region $M \subset \mathbf{R}^n$ with a connected boundary is sufficiently $C^2$-close to a Euclidean one, then it is a minimal filling. By studying the equality case $ \operatorname{vol}(\widetilde M,\tilde g) = \operatorname{vol}(M,g)$ we show that if $ d_{\tilde g}(x,y) = d_g(x,y)$ for all $x,y\in\partial M$ then $(M,g)$ is isometric to $(\widetilde M,\tilde g)$. This gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of Michel’s conjecture.

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Authors

Dmitri Burago

Department of Mathematics
Pennsylvania State University
State College, PA 16802

Sergei Ivanov

V. A. Steklov Mathematical Institute
Russian Academy of Sciences
Fontanka 27
191023 St. Petersburg
Russia