Volume preserving mean curvature flow of revolution hypersurfaces between two equidistants

Cabezas-Rivas, Esther; Miquel, Vicente
January 2012
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p185
Academic Journal
In a rotationally symmetric space $${{\overline M}}$$ around an axis $${\mathcal{A}}$$ (whose precise definition is satisfied by all real space forms), we consider a domain G limited by two equidistant hypersurfaces orthogonal to $${\mathcal{A}}$$. Let $${M \subset {\overline M}}$$ be a revolution hypersurface generated by a graph over $${\mathcal{A}}$$, with boundary in ? G and orthogonal to it. We study the evolution M of M under the volume-preserving mean curvature flow requiring that the boundary of M rests on ? G and stays orthogonal to it. We prove that: (a) the generating curve of M remains a graph; (b) the flow exists as long as M does not touch the rotation axis; (c) under a suitable hypothesis relating the enclosed volume and the area of M, the flow is defined for every $${t\in [0,\infty[}$$ and a sequence of hypersurfaces $${M_{t_n}}$$ converges to a revolution hypersurface of constant mean curvature. Some key points are: (i) the results are true even for ambient spaces with positive curvature, (ii) the averaged mean curvature does not need to be positive and (iii) for the proof it is necessary to carry out a detailed study of the boundary conditions.


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