Uniqueness of a Riemannian metric from least-area data
A classical question in Riemannian geometry is to ask “from what geometric information about the Riemannian manifold can one determine the metric?”. For 2-dimensional, compact, simple manifolds with boundary, Pestov and Uhlmann (2005) proved that one may determine the Riemannian metric from knowledge of the geodesic distance between two boundary points — that is, they showed simple 2-manifolds are “boundary rigid”. Their strategy to show uniqueness for the metric was to relate the boundary distance information to scattering data associated to a harmonic inverse problem.
In this talk, we address a (very broadly speaking) dimension 2 version of the boundary rigidity problem for certain compact, Riemannian 3-manifolds with boundary. We present new results which show that if given any closed curve on the boundary, we know the area of the area-minimizing surface bounded by the curve, then one may uniquely determine the metric. In fact, we prove both a global and local uniqueness result given least-area data for a much smaller class of simple closed curves on the boundary. In our setting, we demonstrate uniqueness for the metric by reformulating parts of our problem as a 2-dimensional inverse problem on a minimal surface. In particular, we relate our minimal area information to knowledge of the Dirichlet-to-Neumann map for the stability operator on a minimal surface.