{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/126935"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/126935","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Convergence of complete Ricci-βat manifolds","abstract":"This thesis is focused on the convergence at inαnity of complete Ricci βat manifolds. In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows.","abstract_html":"This thesis is focused on the convergence at inαnity of complete Ricci βat manifolds. In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows.","abstract_has_math":false,"creators":["Park, Jiewon."],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Tobias Holck Colding."],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-22T22:22:22Z","subjects":["Mathematics."],"languages":["eng"],"rights":["MIT theses may be protected by copyright. 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In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. 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In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/126935"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Convergence of complete Ricci-βat manifolds"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral"]},"updated_at":"2026-07-22T22:22:22Z"}