Determining a Riemannian Metric from Minimal Areas

TL;DR

This paper demonstrates that the Riemannian metric of a 3-ball with mean-convex boundary can be uniquely determined from minimal area data of certain boundary curves, under specific geometric conditions, using advanced mathematical techniques.

Contribution

It introduces a method to recover the metric from minimal area data for a class of 3-manifolds, extending inverse problems in Riemannian geometry.

Findings

Unique determination of the metric from minimal area data under certain conditions.

Local and global results connecting boundary data to interior metric.

Application of pseudo-differential equations and minimal surface theory.

Abstract

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 $\gamma\subset \partial M$. 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 $p$. Additionally, we sketch the proof of a global result with no thin/straight or curvature condition, but assuming the metric admits minimal foliations "from all directions". 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-Calder\'on inverse problem, minimal surface theory, and the careful analysis of a system of pseudo-differential equations.