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An Anisotropic shrinking flow and L_p Minkowski problem
An Anisotropic shrinking flow and L_p Minkowski problem
教师介绍

本讲教师:盛为民
所属学科:理科
人  气:116

课程介绍
Abstract:In this talk, I will introduce my recent work with Caihong Yi on studying anisotropic shrinking flows and the application on L_p Minkowski problem. We consider an shrinking flow of smooth, closed, uniformly convex hypersurfaces in Euclidean R^{n+1} with speed fu^alphasigma_n^{-beta}, where u is the support function of the hypersurface, alpha and beta are two real numbers, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow exists an unique smooth solution for all time and converges smoothly after normalisation to a smooth solution of the equation fu^{alpha-1}sigma_n^{-beta}=c provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on S^n for some cases of alpha and beta. In the case alpha>= 1+nbeta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of fu^{alpha-1}sigma_n^{-beta}=c without any constraint on the initial hypersuface and the function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the L_p Minkowski problem u^{1-p}sigma_n=phi for pin(-n-1,+infty) where phi is a smooth positive function on S^n.

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