https://neckpinch.com/authors/botong-xu/New arXiv papers by Botong Xu, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2103.00842/arxiv:2103.00842Mon, 01 Mar 2021 00:00:00 GMTHaizhong Li, Botong Xu, Ruijia Zhang — In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in ℝⁿ⁺¹ with speed r^(fracαβ)σₖ^(frac1β), where σₖ is the k-th elementary symmetric polynomial of the principal curvatures, α, β are positive constants and r is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of k, α, β. When k≥2, 0<β≤ 1, α≥ β+k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to k-convex. When k ≥ 2, β=k, α≥ 2k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from k=2 to k ≥ 2.Other curvature flowsAnisotropic & nonlocal flowsLong-time existence & convergence