Title：An Anisotropic shrinking flow and L_p Minkowski problem

Speaker：盛为民  教授   （浙江大学）

Time：2019年11月7日（周四）        下午   14:50-15:50

Room：东区管理科研楼  数学科学学院1418室

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^\alpha\sigma_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+n\beta, \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 p\in(-n-1,+\infty) where \phi is a smooth positive function on S^n.