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Aspects of pseudolocality in Ricci flow
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Mcleod, Andrew Donald (2018) Aspects of pseudolocality in Ricci flow. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3352901~S15
Abstract
This thesis explores two separate problems related to the phenomenon of pseudolocality within
Ricci flow.
First, we consider the regularity of noncollapsed three-dimensional Ricci limit spaces via Ricci flow. We introduce a new weakened notion of Ricci flow, termed Pyramid Ricci flow, and use it to establish that a noncollapsed three-dimensional Ricci limit space is homeomorphic to a smooth manifold via a globally-defined homeomorphism that is bi-Hӧlder once restricted to any compact subset. We include the full details of a well-known compactness result which this work relies upon.
Second, we consider the pseudolocality phenomenon in an almost-hyperbolic setting. We obtain an improved pseudolocality result for Ricci flows on two-dimensional surfaces that are initially almost-hyperbolic on large hyperbolic balls. We prove that, at the central point of the hyperbolic ball, the Gauss curvature remains close to the hyperbolic value for a time that grows exponentially in the radius of the ball. This two-dimensional result allows us to precisely conjecture how the phenomenon should appear in the higher dimensional setting.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Ricci flow , Geometry, Hyperbolic, Harmonic maps | ||||
Official Date: | October 2018 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Topping, Peter, 1971- | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | viii, 113 leaves | ||||
Language: | eng |
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