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Mean curvature flow via convex functions on Grassmannian manifolds

Abstract

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the υ-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the υ-function. Under such restrictions, curvature estimates in terms of υ-function composed with the Gauss map can be carried out.

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Author information

Correspondence to Yuanlong Xin.

Additional information

Project supported by the National Natural Science Foundation of China and the Science Foundation of the Ministry of Education of China.

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Xin, Y., Yang, L. Mean curvature flow via convex functions on Grassmannian manifolds. Chin. Ann. Math. Ser. B 31, 315–328 (2010). https://doi.org/10.1007/s11401-009-0173-7

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Keywords

  • Mean curvature flow
  • Convex function
  • Gauss map

2000 MR Subject Classification

  • 53C44