17 research outputs found
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, the
evolving hypersurface M_t meets such tgh ortogonally, 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 discrete along
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.Comment: 24 pages. We have added some lines at the beginning explaining the
notation, and clarified a little bit more the proofs of Proposition 1 and
Theorems 5 and 10, the statements of Proposition 2 and Corollary 3 and an
argument in Remark 1. We have also completed reference 18. A version of this
paper will appear in Mathematische Zeitschrif