{"id":{"repo_id":"toronto-retro","oai_identifier":"oai:utoronto.scholaris.ca:1807/80751"},"canonical_url":"https://search.dev.ndltd.org/etd/toronto-retro/oai:utoronto.scholaris.ca:1807/80751","repository":{"repo_id":"toronto-retro","name":"University of Toronto","base_url":"https://utoronto.scholaris.ca/server/oai/request"},"display":{"title":"Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Balehowsky, Tracey"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Alexakis, Spyros","Nachman, Adrian"],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-11","date_published":"2017-11","updated_at":"2026-07-27T21:28:01Z","subjects":["Area-Minimizing","Geometric Analysis","Inverse Problems","Minimal Surfaces","PDE","Riemannian Geometry"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1807/80751","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Alexakis, Spyros","Nachman, Adrian"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Balehowsky, Tracey"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-11"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-12-19T00:01:43Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-12-19T00:01:43Z"]},{"key":"dc:date.issued","label":"Date","values":["2017-11"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Area-Minimizing","Geometric Analysis","Inverse Problems","Minimal Surfaces","PDE","Riemannian Geometry"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1807/80751"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Alexakis, Spyros","Nachman, Adrian"],"dc:contributor.department":["Mathematics"],"dc:creator":["Balehowsky, Tracey"],"dc:date":["2017-11"],"dc:date.accessioned":["2017-12-19T00:01:43Z"],"dc:date.available":["2017-12-19T00:01:43Z"],"dc:date.issued":["2017-11"],"dc:description.abstract":["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."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1807/80751"],"dc:subject":["Area-Minimizing","Geometric Analysis","Inverse Problems","Minimal Surfaces","PDE","Riemannian Geometry"],"dc:title":["Recovering a Riemannian Metric from Knowledge of the Areas of Properly-Embedded, Area-Minimizing Surfaces"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T21:28:01Z"}