{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8639"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8639","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Calibrations and minimal Lagrangian submanifolds","abstract":"This thesis will be concerned with the geometry of minimal submanifolds in certain Riemannian manifolds which possess some special geometric structure. Those Riemannian manifolds will fall into one of the following categories: 1) A Riemannian manifold M with a calibrating k-form n7. We will derive some intrinsic volume comparison results for calibrated submanifolds of M and give some basic applications to their intrinsic geometry. 2) A Kahler n-fold M with a nowhere vanishing holomorphic (n, 0)-form (we will call'M an almost Calabi-Yau manifold). We will study the geometry of Special Lagrangian submanifolds on M and the global properties of their moduli-space. We will exhibit an example of a compact, simply connected almost Calabi-Yau threefold, which admits a Special Lagrangian torus fibration. We will also show how to construct Special Lagrangian fibrations on non-compact almost Calabi-Yau manifolds using torus actions and give numerous examples of such fibrations. 3) A Kahler-Einstein manifold M with non-zero scalar curvature. We will study the geometry of minimal Lagrangian submanifolds in M and their interaction with the geometry of M. We will also construct some new families of minimal Lagrangian submanifolds in toric Kahler-Einstein manifolds. 4) A Riemannian 7-manifold with holonomy G2. We will construct some new examples of coassociative submanifolds on complete Riemannian 7-manifolds with holonomy G2 via group actions.","abstract_html":"This thesis will be concerned with the geometry of minimal submanifolds in certain Riemannian manifolds which possess some special geometric structure. Those Riemannian manifolds will fall into one of the following categories: 1) A Riemannian manifold M with a calibrating k-form n7. We will derive some intrinsic volume comparison results for calibrated submanifolds of M and give some basic applications to their intrinsic geometry. 2) A Kahler n-fold M with a nowhere vanishing holomorphic (n, 0)-form (we will call'M an almost Calabi-Yau manifold). We will study the geometry of Special Lagrangian submanifolds on M and the global properties of their moduli-space. We will exhibit an example of a compact, simply connected almost Calabi-Yau threefold, which admits a Special Lagrangian torus fibration. We will also show how to construct Special Lagrangian fibrations on non-compact almost Calabi-Yau manifolds using torus actions and give numerous examples of such fibrations. 3) A Kahler-Einstein manifold M with non-zero scalar curvature. We will study the geometry of minimal Lagrangian submanifolds in M and their interaction with the geometry of M. We will also construct some new families of minimal Lagrangian submanifolds in toric Kahler-Einstein manifolds. 4) A Riemannian 7-manifold with holonomy G2. We will construct some new examples of coassociative submanifolds on complete Riemannian 7-manifolds with holonomy G2 via group actions.","abstract_has_math":false,"creators":["Goldstein, Edward, 1977-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Tomasz S. Mrowka."],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001","date_published":"2001","updated_at":"2026-07-22T22:22:27Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/8639","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Tomasz S. Mrowka."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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Those Riemannian manifolds will fall into one of the following categories: 1) A Riemannian manifold M with a calibrating k-form n7. We will derive some intrinsic volume comparison results for calibrated submanifolds of M and give some basic applications to their intrinsic geometry. 2) A Kahler n-fold M with a nowhere vanishing holomorphic (n, 0)-form (we will call'M an almost Calabi-Yau manifold). We will study the geometry of Special Lagrangian submanifolds on M and the global properties of their moduli-space. We will exhibit an example of a compact, simply connected almost Calabi-Yau threefold, which admits a Special Lagrangian torus fibration. We will also show how to construct Special Lagrangian fibrations on non-compact almost Calabi-Yau manifolds using torus actions and give numerous examples of such fibrations. 3) A Kahler-Einstein manifold M with non-zero scalar curvature. We will study the geometry of minimal Lagrangian submanifolds in M and their interaction with the geometry of M. We will also construct some new families of minimal Lagrangian submanifolds in toric Kahler-Einstein manifolds. 4) A Riemannian 7-manifold with holonomy G2. We will construct some new examples of coassociative submanifolds on complete Riemannian 7-manifolds with holonomy G2 via group actions."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Calibrations and minimal Lagrangian submanifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Tomasz S. Mrowka."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. 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We will study the geometry of Special Lagrangian submanifolds on M and the global properties of their moduli-space. We will exhibit an example of a compact, simply connected almost Calabi-Yau threefold, which admits a Special Lagrangian torus fibration. We will also show how to construct Special Lagrangian fibrations on non-compact almost Calabi-Yau manifolds using torus actions and give numerous examples of such fibrations. 3) A Kahler-Einstein manifold M with non-zero scalar curvature. We will study the geometry of minimal Lagrangian submanifolds in M and their interaction with the geometry of M. We will also construct some new families of minimal Lagrangian submanifolds in toric Kahler-Einstein manifolds. 4) A Riemannian 7-manifold with holonomy G2. We will construct some new examples of coassociative submanifolds on complete Riemannian 7-manifolds with holonomy G2 via group actions."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1721.1/8639"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Calibrations and minimal Lagrangian submanifolds"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:27Z"}