An Anisotropic shrinking flow and L_p Minkowski problem

2019/05/12 by Sheng, Weimin, Yi, Caihong
#35K96 (secondary) #53C44 (primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.04679

Abstract

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^alphasigma_n^beta, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on Sⁿ. For the case alpha>= 1+n*beta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of an elliptic equation without any constraint on the initial hypersuface and smooth positive function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the equation of L_p Minkowski problem for p belongs to (-n-1,+infty).

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