Some extensions of the mean curvature flow in Riemannian manifolds

Details Some extensions of the mean curvature flow in Riemannian manifolds Some extensions of the mean curvature flow in Riemannian manifolds

2010-06-27

arxiv.org/abs/1006.5177v3

Mathematics

Differential Geometry

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

Scientific paper

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).

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