Boundary rigidity and volume minimality for almost flat metrics
Abstract. A compact Riemannian manifold with boundary is said to be
boundary rigid if its metric is uniquely determined (up to an isometry) by
the distances between the boundary points; and is said to be a minimal
filling if it has the least volume among all compact Riemannian manifolds
with the same boundary and the same or greater boundary distances.
I will discuss the following result of a recent joint work with D.Burago:
Euclidean regions with Riemannian metrics sufficiently close to a Euclidean
one are minimal fillings and boundary rigid. The proof is based on Gromov's
idea of representing a Riemannian manifold as a surface in an 
-infinity
Banach space, and an observation that the resulting surfaces resemble (in
some sense) minimal surfaces in Euclidean spaces.