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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altschuler, S., Angenent, S., Giga, Y.: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5, 293–358 (1995)
Alikakos, N.D., Freire, A.: The normalized mean curvature flow for a small bubble in a Riemannian manifold. J. Diff. Geom. 64, 247–303 (2003)
Andrews, B.H.: Volume-preserving anisotropic mean curvature flow. Indiana Univ. Math. J. 50, 783–827 (2001)
Angenent, S.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96 (1988)
Athanassenas, M.: Volume-preserving mean curvature flow of rotationally symmetric surfaces. Comment. Math. Helv. 72, 52–66 (1997)
Athanassenas, M.: Behaviour of singularities of the rotationally symmetric, volume-preserving mean curvature flow. Calc. Var. 17, 1–16 (2003)
Cabezas-Rivas, E., Miquel, V.: Volume preserving mean curvature flow in the Hyperbolic Space. Indiana Univ. Math. J. 56(5), 2061–2086 (2007)
Chae, Soo Bong: Lebesgue integration, 2nd edn. Universitext. Springer, New York (1995)
Chen, X.Y., Matano, H.: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimen- sional semilinear heat equations. J. Diff. Equ. 78, 160–190 (1989)
Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991)
Ecker, K.: Regularity theory for mean curvature flow. In: Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhuser Boston, Inc., Boston (2004)
Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proc. A. Math. Soc. 126, 2789–2796 (1998)
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Diff. Geom. 20, 237–266 (1984)
Huisken, G.: Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature. Invent. Math. 84, 463–480 (1986)
Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)
Huisken, B.: Parabolic Monge-Ampere equations on Riemannian manifolds. J. Funct. Anal. 147, 140–163 (1997)
Gray, A.: Tubes, 2nd edn. Birkhuser, Heidelberg (2003)
McCoy, J.A.: Mixed volume preserving curvature flows. Calc. Var. Partial Differ. Equ. 24, 131–154 (2005)
Simon, M.: Mean curvature flow of rotationally symmetric surfaces. B.Sc. Thesis, Dept. of Math. ANU, Canberra
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cabezas-Rivas, E., Miquel, V. Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space. Math. Z. 261, 489–510 (2009). https://doi.org/10.1007/s00209-008-0333-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0333-6