Forced Flow by Powers of the m th Mean Curvature

In this paper, we consider the m th thmean curvature flow of convex hypersurfaces inEuclidean spaces with a general forcing term. Under the assumption that the initial hy-persurface is suitably pinched, we show that the flow may shrink to a point in finite timeif the forcing term is small, or exist for all time and expand to infinity if the forcing termis large enough. The flow can also converge to a round sphere for some special forcingterm and initial hypersurface. Furthermore, the normalization of the flow is carried outso that long time existence and convergence of the rescaled flow are studied. Our workextends Schulze’s flow by powers of the mean curvature and Cabezas-Rivas and Sines-trari’s volume-preserving flow by powers of the m th thmean curvature.