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 C3-smooth, 3-dimensional Riemannian manifold with mean convex boundary ∂M, which is additionally either a) C2-close to Euclidean or b) sufficiently thin, then knowledge of the least areas circumscribed by any simple closed curve γ ⊂ ∂M uniquely determines the metric. In fact, given the least area data for a much more restricted class of curves γ ⊂ ∂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 ∈ ∂M, we prove that knowledge of the least areas circumscribed by any simple closed curve γ in a neighbourhood U ⊂ ∂M of p uniquely determines the metric near p.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2017

Author dc:creator

• Balehowsky, Tracey

Advisors dc:contributor.advisor

• Alexakis, Spyros
• Nachman, Adrian

• Area-Minimizing
• Geometric Analysis
• Inverse Problems
• Minimal Surfaces
• PDE
• Riemannian Geometry

Handle dc:identifier.uri

http://hdl.handle.net/1807/80751

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/80751