The Extension of the H~k Mean Curvature Flow in Riemannian Manifolds

In this paper, the authors consider a family of smooth immersions F _t : M ⁿ → N ⁿ⁺¹ of closed hypersurfaces in Riemannian manifold N ⁿ⁺¹ with bounded geometry, moving by the H ^k mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the H ^k mean curvature flow solution with finite total mean curvature on a finite time interval [0, T _max) can be extended over T _max. This result generalizes the extension theorems in the paper of Li (see “On an extension of the H ^k mean curvature flow, Sci. China Math., 55, 2012, 99–118”).