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


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.


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