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# Asian Journal of Mathematics

## Volume 21 (2017)

### Number 5

### The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action

Pages: 953 – 980

DOI: https://dx.doi.org/10.4310/AJM.2017.v21.n5.a7

#### Author

#### Abstract

In this paper, we study 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, where “almost free” means that the stabilizers of the group action are finite. First we obtain the evolution equations for some geometric quantities along the regularized mean curvature flow. Next, by using the evolution equations, we prove a horizontally strongly convexity preservability theorem for the regularized mean curvature flow. From this theorem, we derive the strongly convexity preservability theorem 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.

#### Keywords

regularized mean curvature flow, horizontally strongly convexity, Riemannian suborbifold

#### 2010 Mathematics Subject Classification

53C42, 53C44

Received 3 August 2015

Accepted 22 April 2016

Published 9 February 2018