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Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M t meets such tgh orthogonally, we prove that: (a) the flow exists while M t does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M t remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-008-0333-6
Publisher site
See Article on Publisher Site

Abstract

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M t meets such tgh orthogonally, we prove that: (a) the flow exists while M t does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M t remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.

Journal

Mathematische Zeitschrift – Springer Journals

Published: Mar 18, 2008

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