160 research outputs found
Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space
This paper concerns closed hypersurfaces of dimension in the
hyperbolic space of constant sectional curvature
evolving in direction of its normal vector, where the speed is given
by a power of the th mean curvature plus a volume
preserving term, including the case of powers of the mean curvature and of the
\mbox{Gau\ss} curvature. The main result is that if the initial hypersurface
satisfies that the ratio of the biggest and smallest principal curvature is
close enough to 1 everywhere, depending only on , , and ,
then under the flow this is maintained, there exists a unique, smooth solution
of the flow for all times, and the evolving hypersurfaces exponentially
converge to a geodesic sphere of , enclosing the
same volume as the initial hypersurface.Comment: 36page
MEAN CURVATURE FLOW OF SUBMANIFOLDS WITH SMALL TRACELESS SECOND FUNDAMENTAL FORM
Consider a family of smooth immersions F(; t) : Mn Mn+k of submanifolds in Mn+k moving by mean curvature flow = , where is the mean curvature vector for the evolving submanifold. We prove that for any n >-2 and k>-1, the flow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in finite time, and the corresponding normalized flow converges exponentially in the C-topology, to an n-sphere in some subspace Mn+1 of Mn+k
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