# Forced Flow by Powers of the $\emph{m}^{\th}$ Mean Curvature

Wu, Chuanxi and Tian, Daping and Li, Guanghan (2010) Forced Flow by Powers of the $\emph{m}^{\th}$ Mean Curvature. Armenian Journal of Mathematics, 3 (2). pp. 61-91. ISSN 1829-1163

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## Abstract

In this paper, we consider the $m^{\th}$ 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 $m^{\th}$ mean curvature.

Item Type: Article 35-xx Partial differential equations53-xx Differential geometry 255 Professor Anry Nersesyan 17 Jun 2010 12:34 19 Apr 2011 02:24

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