Determining a Riemannian Metric from Minimal Areas
| Autor: | Spyros Alexakis, Tracey Balehowsky, Adrian I. Nachman |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Minimal surface General Mathematics 010102 general mathematics Boundary (topology) Fundamental theorem of Riemannian geometry Isometry (Riemannian geometry) Topology 01 natural sciences Manifold Statistical manifold Mathematics - Analysis of PDEs Differential Geometry (math.DG) 0103 physical sciences Metric (mathematics) FOS: Mathematics 010307 mathematical physics 0101 mathematics Fisher information metric Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1711.09379 |
| Popis: | 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. Comment: Second result optimized to a broader class of thin/straight manifolds. 66 pages, 8 figures |
| Databáze: | OpenAIRE |
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