Cabezas-Rivas, Esther and Miquel, Vicente (2009) Volume-preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space. Mathematische Zeitschrift, Vol.261 (No.3). pp. 489-510. doi:10.1007/s00209-008-0333-6 ISSN 0025-5874.

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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.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Mathematische Zeitschrift
Publisher:	Springer
ISSN:	0025-5874
Official Date:	2009
Dates:	Date Event 2009 Published
Volume:	Vol.261
Number:	No.3
Page Range:	pp. 489-510
DOI:	10.1007/s00209-008-0333-6
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

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