Miglioranza, Mattia;
(2020)
The volume preserving mean curvature flow in a compact Riemannian manifold.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
In this thesis we investigate the volume preserving mean curvature flow (VPMCF) of a closed and convex hypersurface M inside of a compact Riemannian manifold N. When the ambient manifold is the Euclidean space, long time existence and convergence of the solution to a sphere have been already proved. In the general Riemannian case, this approach cannot be readily generalised, because of the interaction between the evolving hypersurface and the geometry of the ambient space. Alikakos and Freire overcome these difficulties, using although an infinite-dimensional dynamical systems approach and results from semigroup theory. In our work, instead, we offer a classical and more geometric outlook. We therefore exploit the isoperimetric nature of the flow: the hypersurface M is in fact moving inside N in a way to keep the volume of the region it encloses xed, while its area is strictly decreasing. Thanks to this isoperimetric characteristic, we prove that, if the initial hypersurface is close enough to a small geodesic ball in N (a bubble), it keeps itself close even at the final existence time T (short time existence). The last fact, combined with good estimates of the major geometric quantities of M, allows us to extend the flow indefinitely for all times (immortal flow) and therefore to study its asymptotic behaviour. This is quite interesting, since, except for special cases, geodesic spheres are not equilibria for the VPMCF and, in general, the existence of time independent solutions is a non trivial issue. We conclude our work by studying the asymptotic behaviour of a solution of the VPMCF. We prove that there exists at least a subsequence of times such that a subsequence of the family of bubbles converges to a limit surface of constant mean curvature.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | The volume preserving mean curvature flow in a compact Riemannian manifold |
| Event: | UCL (University College London) |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10094388 |
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