An anisotropic shrinking flow and \(L_p\) Minkowski problem
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Publication:6159281
DOI10.4310/CAG.2022.V30.N7.A3zbMATH Open1520.53089arXiv1905.04679MaRDI QIDQ6159281
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Publication date: 1 June 2023
Published in: (Search for Journal in Brave)
Abstract: We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_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 S^n. 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).
Full work available at URL: https://arxiv.org/abs/1905.04679
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