Determining a Riemannian Metric from Minimal Areas

Understand

We prove that if $(M,g)$ is a topological 3-ball with a $C^4$-smooth Riemannian metric $g$, and mean-convex boundary $\partial M$ then knowledge of least areas circumscribed by simple closed curves $\gamma \subset \partial M$ 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.
• %sufficiently thin.
• In fact, the least area data that we require is for a much more restricted class of curves $\gamma\subset \partial M$.