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Fine analysis of mean curvature flow through singularities
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Daniels-Holgate, Joshua (2023) Fine analysis of mean curvature flow through singularities. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3982139~S15
Abstract
We provide a short survey on the history of the mean curvature flow and the theory of flow through singularities. We establish the existence of a smooth flow with surgery approximating weak mean curvature flows with only spherical and neck pinch singularities, thereby dropping the standard global 2-convexity assumption.
This makes use of the resolution of the mean-convex neighbourhood conjecture of Choi–Haslhofer–Hershkovits, and Choi–Haslhofer–Hershkovits–White and a barrier argument for flows with surgery. We conclude our discussion of surgery by utilising the surgery flow, in combination with results of Choi–Chodosh–Mantoulidis–Schulze for generic flows, to increase the known entropy bound for the Schoenflies conjecture in R4. We then consider mean curvature flow of compact hypersurfaces through conical singularities. We demonstrate a uniqueness theorem for flows with tangent flows modelled on the flow generated by a smooth, stable expander with a linearly growing Jacobi field. Moreover, we demonstrate the forward tangent flow at the conical singularity of the outer-most Brakke flows are modelled on the outer-most expanders of the cone, when said expanders are smooth. Combined with work of Ilmanen–White, this demonstrates genus drop for the outer-most flows through such singularities, answering a conjecture of Ilmanen. Finally, we deduce the following dichotomy in dimensions 2 ≤ n ≤ 6: The flow from a compact hypersurface with isolated conical singularity fattens if and only if the flow from the model cone fattens.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Flows (Differentiable dynamical systems), Curvature, Singularities (Mathematics), Geometric analysis | ||||
Official Date: | September 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Schulze, Felix | ||||
Sponsors: | University of Warwick. Mathematics Institute | ||||
Format of File: | |||||
Extent: | vi, 118 pages | ||||
Language: | eng |
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