Convexity and gradient estimates for fully nonlinear curvature flows

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dc.contributor.advisor Huisken, Gerhard (Prof. Dr.)
dc.contributor.author Lynch, Stephen
dc.date.accessioned 2020-11-20T10:18:59Z
dc.date.available 2020-11-20T10:18:59Z
dc.date.issued 2020-11-20
dc.identifier.other 1740232372 de_DE
dc.identifier.uri http://hdl.handle.net/10900/109737
dc.identifier.uri http://nbn-resolving.de/urn:nbn:de:bsz:21-dspace-1097373 de_DE
dc.identifier.uri http://dx.doi.org/10.15496/publikation-51113
dc.description.abstract We study deformations of hypersurfaces with normal velocity given by a smooth symmetric increasing function of the principal curvatures. Specifically we study flows where the speed is a nonlinear concave function, so that at the coordinate level the evolution is governed by a fully nonlinear parabolic PDE. For each $k \geq 3$ we construct the first flows of this kind which smoothly deform any compact $k$-convex hypersurface of Euclidean space through a family of hypersurfaces which are also $k$-convex, before forming finite-time singularities which are necessarily convex (by $k$-convexity we mean that the sum of the smallest $k$ principal curvatures is everywhere positive). That is, we show that $k$-convexity is preserved and establish an analogue of the Huisken-Sinestrari convexity estimate, which implies convexity of singularities for mean-convex mean curvature flow. In contrast to the mean curvature flow, the fully nonlinear flows constructed here also preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature is suitably pinched. We then employ our convexity estimate to prove Harnack and derivative estimates for the second fundamental form of solutions which are embedded. These results imply for example that sequences of rescalings about a singularity satisfy universal bounds for the second fundamental form and all of its higher derivatives on compact subsets of spacetime. The estimates are obtained by generalising an induction on scales technique introduced by Brendle-Huisken for two-convex flows to the $k$-convex setting. Our arguments apply to a general class of flows including mean-convex mean curvature flow, and in this case we recover the influential global Harnack inequality of Haslhofer-Kleiner, but without using Huisken's monotonicity formula. en
dc.language.iso en de_DE
dc.publisher Universität Tübingen de_DE
dc.rights ubt-podok de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=de de_DE
dc.rights.uri http://tobias-lib.uni-tuebingen.de/doku/lic_mit_pod.php?la=en en
dc.subject.classification Geometrie de_DE
dc.subject.ddc 510 de_DE
dc.subject.other mean curvature flow en
dc.subject.other fully nonlinear curvature flows en
dc.subject.other curvature en
dc.subject.other hypersurfaces en
dc.subject.other partial differential equations en
dc.title Convexity and gradient estimates for fully nonlinear curvature flows en
dc.type PhDThesis de_DE
dcterms.dateAccepted 2020-11-13
utue.publikation.fachbereich Mathematik de_DE
utue.publikation.fakultaet 7 Mathematisch-Naturwissenschaftliche Fakultät de_DE

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