Mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces

Naoyuki Koike

doi:10.4067/S0719-06462018000300013

DOI:

https://doi.org/10.4067/S0719-06462018000300013

Abstract

In this paper, we investigate the mean curvature flows starting from all leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in C^âˆž-topology. These results give the geometric information between the leaves.

Keywords

error function based activation function , multivariate quasi-interpolation neural network approximation , Kantorovich-Shilkret type operator

Naoyuki Koike Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka Shinjuku-ku, Tokyo 162-8601, Japan.

Pages: 13–29
Date Published: 2019-03-15
Vol. 20 No. 3 (2018)

M. M. Alexandrino and M. Radeschi, Mean curvature flow of singular Riemannian foliations, J. Geom. Anal. 26 2204–2220 (2015).

B. Andrews and C. Baker, Mean curvature flow of pinched submanifolds to spheres, J. Differential Geom. 85 (2010) 357-396.

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20(1984) 237-266.

G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. math. 84 (1986) 463-480.

N. Koike, Examples of a complex hyperpolar action without singular orbit, Cubo A Math. J.12 (2010) 131-147.

N. Koike, Collapse of the mean curvature flow for equifocal submanifolds, Asian J. Math. 15(2011) 101-128.

N. Koike, Collapse of the mean curvature flow for isoparametric submanifolds in a symmetric space of non-compact type, Kodai Math. J. 37 (2014) 355-382.

J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329.