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In this paper, we study the regularized mean curvature flow starting frominvariant hypersurfaces in a Hilbert space equipped with an isometric almostfree Hilbert Lie group action whose orbits are minimal regularizablesubmanifolds, where "almost free" means that the stabilizers of the groupaction are finite. First we obtain the evolution equations for some geometricquantities along the regularized mean curvature flow. Next, by using theevolution equations, we prove a horizontally strongly convexity preservabilitytheorem for the regularized mean curvature flow. From this theorem, we derivethe strongly convexity preservability theorem for the mean curvature flowstarting from compact Riemannian suborbifolds in the orbit space (which is aRiemannian orbifold) of the Hilbert Lie group action.

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