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Mean curvature flow with free boundary outside a hypersphere


Authors: Glen Wheeler and Valentina-Mira Wheeler
Journal: Trans. Amer. Math. Soc. 369 (2017), 8319-8342
MSC (2010): Primary 53C44, 58J35
DOI: https://doi.org/10.1090/tran/7305
Published electronically: September 7, 2017
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Abstract: The purpose of this paper is twofold: first, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and second, to illustrate the application of Killing vector fields in the preservation of graphicality for the mean curvature flow with free boundary. To this end we focus on the mean curvature flow of a topological annulus with inner boundary meeting a standard $ n$-sphere in $ \ensuremath {\mathbb{R}}^{n+1}$ perpendicularly and outer boundary fixed to an $ (n-1)$-sphere with radius $ R>1$ translated by a vector $ he_{n+1}$ for $ h\in \ensuremath {\mathbb{R}}$ where $ \{e_i\}_{i=1,\ldots ,n+1}$ is the standard basis of $ \ensuremath {\mathbb{R}}^{n+1}$. We call this the sphere problem. Our work is set in the context of graphical mean curvature flow with either symmetry or mean concavity/convexity restrictions. For rotationally symmetric initial data we obtain, depending on the exact configuration of the initial graph, either long time existence and convergence to a minimal hypersurface with boundary or the development of a finite-time curvature singularity. With reflectively symmetric initial data we are able to use Killing vector fields to preserve graphicality of the flow and uniformly bound the mean curvature pointwise along the flow. Finally we prove that the mean curvature flow of an initially mean concave/convex graphical surface exists globally in time and converges to a piece of a minimal surface.


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Additional Information

Glen Wheeler
Affiliation: Institute for Mathematics and its Applications, University of Wollongong, Northfields Avenue, Wollongong, New South Wales 2522, Australia
Email: glenw@uow.edu.au

Valentina-Mira Wheeler
Affiliation: Institute for Mathematics and its Applications, University of Wollongong, Northfields Avenue, Wollongong, New South Wales 2522, Australia
Email: vwheeler@uow.edu.au

DOI: https://doi.org/10.1090/tran/7305
Keywords: Mean curvature flow, free boundary conditions, geometric analysis
Received by editor(s): May 30, 2014
Published electronically: September 7, 2017
Additional Notes: The first author was supported by Alexander-von-Humboldt Fellowship 1137814 and the second author by DFG Grant ME 3816/2-1. During the completion of the work the authors were supported by ARC Discovery Project DP120100097 at the University of Wollongong. The second author is the corresponding author.
Article copyright: © Copyright 2017 American Mathematical Society