Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces

In this thesis, we prove that if $(M,g)$ is a $C^3$-smooth, 3-dimensional Riemannian manifold with mean convex boundary $\partial M$, which is additionally either a) $C^2$-close to Euclidean or b) sufficiently thin, then knowledge of the least areas circumscribed by any simple closed curve $\gamma \subset \partial M$ uniquely determines the metric. In fact, given the least area data for a much more restricted class of curves $\gamma\subset \partial M$, we uniquely determine the metric. We also prove a corresponding local result: assuming only that $(M,g)$ has strictly mean convex boundary at a point $p\in\partial M$, we prove that knowledge of the least areas circumscribed by any simple closed curve $\gamma$ in a neighbourhood $U\subset \partial M$ of $p$ uniquely determines the metric near $p$.Ph.D