Forced Flow by Powers of the $\textit{m}^{{\rm th}}$ Mean Curvature

Authors

  • Chuanxi Wu School of Mathematics and Computer Science, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, P. R. China
  • Daping Tian School of Mathematics and Computer Science, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, P. R. China
  • Guanghan Li School of Mathematics and Computer Science, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, P. R. China

Abstract

In this paper, we consider the $m^{{\rm th}}$ mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. Under the assumption that the initial hypersurface is suitably pinched, we show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Schulze's flow by powers of the mean curvature and Cabezas-Rivas and Sinestrari's volume-preserving flow by powers of the $m^{{\rm th}}$ mean curvature.

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Published

2010-06-17

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Articles

How to Cite

Forced Flow by Powers of the $\textit{m}^{{\rm th}}$ Mean Curvature. (2010). Armenian Journal of Mathematics, 3(2), 61-91. http://test.armjmath.sci.am/index.php/ajm/article/view/70