In this talk, we address the following question: Given any simple closed curve $\gamma$ on the boundary of a Riemannian 3-manifold $(M,g)$, suppose the area of the least-area surfaces bounded by $\gamma$ are known. From this data may we uniquely recover $g$?

In several settings, we show the the answer is yes. In fact, we prove both global and local uniqueness results given least-area data for a much smaller class of curves on the boundary. We demonstrate uniqueness for $g$ by reformulating parts of the problem as a 2-dimensional inverse problem on an area-minimizing surface. In particular, we relate our least-area information to knowledge of the Dirichlet-to-Neumann map for the stability operator on a minimal surface.

Broadly speaking, the question we address is a dimension 2 version of the classical boundary rigidity problem for simply connected, Riemannian 3-manifolds with boundary, in which one seeks to determine $g$ given the distance between any two points on the boundary. We will also briefly review this problem of boundary rigidity as it relates to aspects of our question of recovering $g$ from knowledge of areas.

This is joint work with S. Alexakis and A. Nachman.