Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.767912
Title: Structure of singular sets local to cylindrical singularities for stationary harmonic maps and mean curvature flows
Author: Wells-Day, Benjamin Michael
ISNI:       0000 0004 7651 549X
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2019
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Abstract:
In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.
Supervisor: Wickramasekera, Neshan Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.767912  DOI:
Keywords: harmonic maps ; mean curvature flows ; brakke flows ; singularities ; geometric measure theory
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