Mean Curvature Flow of Singular Riemannian Foliations

Abstract

Abstract. Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. These results generalize previous results of Liu and Terng, Pacini and Koike. In particular our results can be applied to partitions of Riemannian manifolds into orbits of actions of compact groups of isometries.

Lie Groups and Geometric Aspects of Isometric Actions

“…For example, it is proved that focal submanifolds of an isoparametric function are minimal, and that there exists at least one minimal hypersurface level set. These results are also analogous to properties of orbits of a cohomogeneity one action, see Remark 6.43, and can be alternatively deduced as an application of the mean curvature flow on polar foliations, see [17].…”

“…where M/G is equipped with the natural orbit metric, whose associated distance is the above orbital distance. 17 We conclude this section proving the so-called Principal Orbit Theorem, which guarantees that the subset of points in M on principal orbits is open and dense, and that M/G has an open and dense subset which is a smooth manifold (see (iii) If G(x) and G(y) are principal orbits, there exists g ∈ G such that G x = gG y g −1 .…”

“…This illustrates the fact that the metrics g t in (6.27) must satisfy appropriate conditions at the corresponding endpoints t → ±1 17. This is a horizontal curve in the sense that it is orthogonal to all circle orbits.…”

“…An Alexandrov space is a finite-dimensional length metric space with a lower curvature bound, in a comparison geometry sense. For details, see[59,107] 17. More generally, if μ is not free, then π : M → M/G is a submetry, which is a generalization of submersions to metric spaces.…”

Algebraic nature of singular Riemannian foliations in spheres

“…Note that κ is a smooth form on M 0 . But be aware, that κ is definitely non-smooth at singular leaves (indeed, H 2 explodes quadratically as one approaches a singular point, [5,Prop. 4.3]).…”