https://neckpinch.com/authors/guangming-hu/New arXiv papers by Guangming Hu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2411.06274arxiv:2411.06274Sat, 09 Nov 2024 00:00:00 GMTGuangming Hu, Ziping Lei, Yanlin Li, Hao Yu — In 1991, Beardon and Stephenson [2] generalized the classical Schwarz-Pick lemma in hyperbolic geometry to the discrete Schwarz-Pick lemma for Andreev circle packings. This paper continues to investigate the discrete Schwarz-Pick lemma for generalized circle packings (including circle, horocycle or hypercycle) in hyperbolic background geometry. Since the discrete Schwarz-Pick lemma is to compare some geometric quantities of two generalized circle packings with different boundary values, we first show the existence and rigidity of generalized circle packings with boundary values, and then we introduce the method of combinatorial Calabi flows to find the generalized circle packings with boundary values. Moreover, motivated by the method of He [21], we propose the maximum principle for generalized circle packings. Finally, we use the maximum principle to prove the discrete Schwarz-Pick lemma for generalized circle packings.Other curvature flowsKähler geometryStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2407.08496arxiv:2407.08496Thu, 11 Jul 2024 00:00:00 GMTGuangming Hu, Sicheng Lu, Dong Tan, Youliang Zhong, Puchun Zhou — This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully characterize the sufficient and necessary conditions and show the uniqueness. Furthermore, we introduce the combinatoral Ricci flow to find the desired degenerated circle packed surface, analougus to the methods of Chow-Luo and Takatsu.Ricci flowStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2404.05982arxiv:2404.05982Tue, 09 Apr 2024 00:00:00 GMTGuangming Hu, Ziping Lei, Yu Sun, Puchun Zhou — In this paper, we study the existence and rigidity of (degenerated) circle pattern metric with prescribed total geodesic curvatures in spherical background geometry. To find the (degenerated) circle pattern metric with prescribed total geodesic curvatures, we define some prescribed combinatorial Ricci flows and study the convergence of flows for (degenerated) circle pattern metrics. We solve the prescribed total geodesic curvature problem and provide two methods to find the degenerated circle pattern metric with prescribed total geodesic curvatures. As far as we know, this is the first degenerated result for total geodesic curvatures in spherical background geometry.Ricci flowStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2403.05777arxiv:2403.05777Sat, 09 Mar 2024 00:00:00 GMTGuangming Hu, Ziping Lei, Yi Qi, Puchun Zhou — In hyperbolic background geometry, we investigate a generalized circle packing (including circles, horocycles and hypercycles) with conical singularities on a surface with boundary, which has a total geodesic curvature on each generalized circle of this circle packing and a discrete Gaussian curvature on the center of each dual circle. The purpose of this paper is to find this type of circle packings with prescribed total geodesic curvatures on generalized circles and discrete Gaussian curvatures on centers of dual circles. To achieve this goal, we firstly establish existence and rigidity on this type of circle packings by the variational principle. Secondly, for p>1, we introduce combinatorial p-th Calabi flows to find the circle packing with prescribed total geodesic curvatures on generalized circles and discrete Gaussian curvatures on centers of dual circles for the first time.Other curvature flowsSingularitiesSurfacesKähler geometryStability & uniquenessKähler–Einstein & K-stabilityDiscrete & graph Ricci https://arxiv.org/abs/2311.10528arxiv:2311.10528Fri, 17 Nov 2023 00:00:00 GMTGuangming Hu, Yi Qi, Yu Sun, Puchun Zhou — This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the 1-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdière's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.Ricci flowSingularitiesSurfacesStability & uniquenessDiscrete & graph Ricci