Ricci flow : curvature concentration and Sobolev bounds
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Martens, Adam
issued: 2025-06-13T22:17:19Z
issued: 2025
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: In this dissertation, we study the Ricci flow and its applications on manifolds that have small enough scale-invariant integral curvature (so-called "curvature concentration") relative to its Sobolev bounds. Our primary results are summarized in the following three items. (1) We show that if the bounds are assumed to be global for a bounded curvature manifold, then the Ricci flow emanating from this initial data exists for all time, with faster than critical curvature decay. We also show topological rigidity assuming that the curvature is globally sub-critically integrable - an assumption that is, for example, satisfied by the well-studied family of "asymptotically flat" manifolds. (2) We sharpen a gap theorem of Chan and Lee for nonnegative Ricci curvature manifolds (possibly unbounded curvature) that have positive asymptotic volume ratio (in the nonnegative Ricci curvature setting, we show that volume bounds are equivalent to Sobolev bounds) and small enough curvature concentration, by showing that the curvature concentration need only depend linearly on the asymptotic volume ratio. We prove the result by exhibiting a long-time Ricci flow solution with faster than critical curvature decay (item (1) above is used as one element of the proof), which allows us to shift the limiting contradiction argument to time infinity and thus obtain an explicit bound on the size of the gap. (3) In recent work of Chan-Huang-Lee, it is shown that if a (possibly unbounded curvature) manifold enjoys uniform bounds on the negative part of the scalar curvature, the local entropy, and volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially. We show that the bound on scalar curvature assumption above is redundant. We also give some applications of this quantitative short-time existence, including Ricci flow smoothing for measure space limits, a Gromov-…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/91357
note: Ricci flow: curvature concentrationand Sobolev boundsbyAdam MartensB.Sc., The University of British Columbia, 2019M.Sc., The University of British Columbia, 2021A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2025© Adam Martens 2025The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Ricci flow: curvature concentration and Sobolev boundssubmitted by Adam Martens in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Mathematics.Examining Committee:Albert Chau, Professor, Mathematics, UBCSupervisorAilana Fraser, Professor, Mathematics, UBCSupervisory Committee MemberSe´bastien Picard, Assistant Professor, Mathematics, UBCSupervisory Committee MemberStephen Gustafson, Professor, Mathematics, UBCUniversity ExaminerMark Van Raamsdonk, Professor, Physics and Astronomy, UBCUniversity ExaminerRichard Bamler, Professor, Mathematics, University of California, BerkeleyExternal ExamineriiAbstractIn this dissertation, we study the Ricci flow and its applications on man-ifolds that have small enough scale-invariant integral curvature (so-called“curvature concentration”) relative to its Sobolev bounds. Our primaryresults are summarized in the following three items.(1) We show that if the bounds are assumed to be global for a boundedcurvature manifold, then the Ricci flow emanating from this initial dataexists for all time, with faster than critical curvature decay. We also showtopological rigidity assuming that the curvature is globally sub-criticallyintegrable - an assumption that is, for example, satisfied by the well-studiedfamily of “asymptotically flat” manifolds.(2) We sharpen a gap theorem of Chan and Lee for nonnegative Ricci cur-vature manifolds (possibly unbounded curvature) …
has type: Thesis/Dissertation
date issued: 2025-11
is shown at: 10.14288/1.0449119
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Chau, Albert
type: Text
identifier URI: http://hdl.handle.net/2429/91357