The Regularized Mean Curvature Flow for Invariant Hypersurfaces in a Hilbert Space
In this note, I state some results for the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds. First we derive the evolution equations for some geometric quantities along this flow. Some of the evolution equations are described by using the O’Neill fundamental tensor of the orbit map of the Hilbert Lie group action, where we note that the O’Neill fundamental tensor implies the obstruction for the integrability of the horizontal distribution of the orbit map. Next, by using the evolution equations, we derive some results for this flow. Furthermore, we derive some results for the mean curvature flow starting from compact Riemannian suborbifolds in the orbit space (which is a Riemannian orbifold) of the Hilbert Lie group action.
KeywordsCurvature Flow Curvature Vector Riemannian Connection Fundamental Tensor Hilbert Space Versus
The author is grateful to JSPS for support (Grant-in-Aid for Science Research (C), no.25400076).
- 4.Heintze, E., Liu, X., Olmos, C.: Isoparametric submanifolds and a Chevalley-type restriction theorem, integrable systems, geometry, and topology. In: AMS/IP Studies in Advanced Mathematics, vol. 36, pp. 151–190. American Mathematical Society, Providence (2006)Google Scholar
- 8.Koike, N.: The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action. arXiv:1210.2539v2 [math.DG]Google Scholar
- 13.Thurston, W.P.: Three-dimensional Geometry and Topology. Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)Google Scholar