  In this paper, we investigate a regularized mean curvature flow starting from
an invariant hypersurface in a Hilbert space equipped with an isometric and
almost free action of a Hilbert Lie group whose orbits are minimal
regularizable submanifolds. We prove that, if the initial invariant
hypersurface satisfies a certain kind of horizontally convexity condition and
some additional conditions, then it collapses to an orbit of the Hilbert Lie
group action along the regularized mean curvature flow. In the final section,
we state a vision for applying the study of the regularized mean curvature flow
to the gauge theory.

Differential Geometry