https://neckpinch.com/authors/guanghan-li/New arXiv papers by Guanghan Li, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2601.14659arxiv:2601.14659Wed, 21 Jan 2026 00:00:00 GMTGuanghan Li, Chenyang Liu — In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.Other curvature flowsGauss curvature flowFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2209.04601arxiv:2209.04601Sat, 10 Sep 2022 00:00:00 GMTShanwei Ding, Guanghan Li — In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the L^p dual Minkowski problem from the dual point of view. The results of corresponding L^p versions are L^p dual Minkowski problem for p>q; and even L^p dual Minkowski problem for p>−1, or q<1, or some ranges of p<0<q, which contain all existence results for smooth measures up to now except p=q or q=n+1 (L^p Minkowski problem).Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsLong-time existence & convergence https://arxiv.org/abs/2207.03114arxiv:2207.03114Thu, 07 Jul 2022 00:00:00 GMTShanwei Ding, Guanghan Li — In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE G(x,u_K,Du_K)F(D²u_K+u_KI)=1 on Sⁿ for a convex body K, where D is the covariant derivative with respect to the standard metric on Sⁿ and I is the unit matrix of order n. This result covers many previous known solutions to L^p dual Minkowski problem, L^p dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc.. Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).Other curvature flowsGauss curvature flowAnisotropic & nonlocal flowsHomogeneous spacesLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2203.02165arxiv:2203.02165Fri, 04 Mar 2022 00:00:00 GMTShanwei Ding, Guanghan Li — In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space ℝⁿ⁺¹ with speed ψu^αρ^δf^(−β), where ψ is a smooth positive function on unit sphere, u is the support function of the hypersurface, ρ is the radial function, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When ψ=1, we prove that the flow exists for all time and converges to infinity if α+δ+β≤1, β>0 and α≤0, while in case α+δ+β>1,α,δ≤0, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered the origin. In particular, the results of Gerhardt and Urbas can be recovered by putting α=δ=0. Our previous works can be recovered by putting δ=0. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to L^p-Minkowski problem and L^p-Christoffel-Minkowski problem with constant prescribed data. Similarly, we pose the L^p dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to L^p dual Minkowski problem and L^p dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the longtime existence and convergence of a class of anisotropic flows (i.e. for general function ψ). The final result not only gives a new proof of many previously known solutions to L^p dual Minkowski problem, L^p-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to L^p dual Christoffel-Minkowski problem with some conditions.Other curvature flowsInverse mean curvature flowGauss curvature flowAnisotropic & nonlocal flowsSingularitiesStability & uniquenessLong-time existence & convergence