Abstract
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”).
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Project supported by the National Natural Science Foundation of China (Nos. 11301399, 11126189, 11171259, 11126190), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120141120058), the China Postdoctoral Science Foundation (No. 20110491212) and the Fundamental Research Funds for the Central Universities (Nos. 2042011111054, 20420101101025).
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Qiu, H., Ye, Y. & Zhu, A. The extension of the H k mean curvature flow in Riemannian manifolds. Chin. Ann. Math. Ser. B 35, 191–208 (2014). https://doi.org/10.1007/s11401-014-0827-y
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DOI: https://doi.org/10.1007/s11401-014-0827-y