Some extensions of the mean curvature flow in Riemannian manifolds

Abstract

Given a family of smooth immersions $F_t:M^n\to N^{n+1}$ of closed hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded geometry, moving by the mean curvature flow, we show that at the first finite singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a recent result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of N. Le in the Euclidean case (arXiv: math.DG/1002.4669v2).

arXiv comment18 pages. Theorem 1.6 was improved. The proof of Corollary 1.7 was corrected

References in corpus (4)

• On Type I Singularities in Ricci flow
• The mean curvature at the first singular time of the mean curvature flow
• The extension and convergence of mean curvature flow in higher codimension
• Remarks on curvature behavior at the first singular time of the Ricci flow