Determining a Riemannian metric from minimal areas

We prove that if (M,g) is a topological 3-ball with a C-4-smooth Riemannian metric g, and mean-convex boundary am, then knowledge of least areas circumscribed by simple closed curves gamma subset of partial derivative M uniquely determines the metric g, under some additional geometric assumptions. These are that g is either a) C-3-close to Euclidean or b) satisfies much weaker geometric conditions which hold when the manifold is to a sufficient degree either thin, or straight. In fact, the least area data that we require is for a much smaller subset of curves gamma subset of partial derivative M. We also prove a local result: Given a point p is an element of partial derivative M where partial derivative M is mean convex, we provide a reconstruction of the metric near p, given suitable area data near p only. The proofs rely on finding the metric along a continuous sweep-out of M by area-minimizing surfaces; they bring together ideas from the 2D-Calderon inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations. (C) 2020 Published by Elsevier Inc.