https://neckpinch.com/authors/puchun-zhou/New arXiv papers by Puchun Zhou, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2506.05036arxiv:2506.05036Thu, 05 Jun 2025 00:00:00 GMTHuabin Ge, Bobo Hua, Hao Yu, Puchun Zhou — In his seminal work, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem. In this paper, we introduce the combinatorial Ricci flow for infinite ideal circle patterns, a discrete analogue of Ricci flow on non-compact Riemannian manifolds, and prove a characterization of such circle patterns under certain combinatorial conditions. Our results provide affirmative solutions to Rivin's problem.Ricci flow3-manifoldsBoundaries & non-compactDiscrete & graph Ricci https://arxiv.org/abs/2505.20091arxiv:2505.20091Mon, 26 May 2025 00:00:00 GMTXinrong Zhao, Puchun Zhou — In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce a prescribed curvature flow-a discrete analogue of the Ricci flow on noncompact surfaces-specifically adapted to the setting of infinite cellular decompositions. We establish the well-posedness of the flow and prove two convergence results under certain conditions. Our approach resolves the prescribed total geodesic curvature problem for a broad class of surfaces with infinite cellular decompositions, yielding, in certain cases, smooth hyperbolic surfaces of infinite topological type with geodesic boundaries or cusps. Moreover, the proposed flow provides a method for constructing hyperbolic metrics from appropriate initial data.Ricci flowOther curvature flowsSurfacesBoundaries & non-compactDiscrete & graph Ricci https://arxiv.org/abs/2504.05817arxiv:2504.05817Tue, 08 Apr 2025 00:00:00 GMTHuabin Ge, Bobo Hua, Puchun Zhou — In this paper, we introduce combinatorial Ricci flows (CRFs in short) in Euclidean and hyperbolic background geometries on infinite triangulations of the open disk, which are discrete analogs of Ricci flows on simply connected open surfaces. We establish well-posedness results, the existence and the uniqueness, of CRFs in both Euclidean and hyperbolic background geometries. Moreover, we prove convergence results of CRFs, which indicate a uniformization theorem for CRFs on infinite disk triangulations. As an application, we prove an existence result of circle-packing metrics with infinite prescribed cone angles in hyperbolic background geometry. To our knowledge, these are the first results of CRFs on infinite triangulations.Ricci flowSurfacesStability & 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 https://arxiv.org/abs/2305.19611arxiv:2305.19611Wed, 31 May 2023 00:00:00 GMTZiping Lei, Puchun Zhou — Combinatorial Calabi flows are introduced by Ge in his Ph.D. thesis (Combinatorial methods and geometric equations, Peking University, Beijing, 2012), and have been studied extensively in Euclidean and hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow in spherical background geometry for finding ideal circle patterns with prescribed total geodesic curvatures. We prove that the solution of combinatorial Calabi flow exists for all time and converges if and only if there exists an ideal circle pattern with prescribed total geodesic curvatures. We also show that if it converges, it will converge exponentially fast to the desired metric, which provides an effective algorithm to find certain ideal circle patterns. To our knowledge, it is the first combinatorial Calabi flow in spherical background geometry.Other curvature flowsKähler geometryLong-time existence & convergence https://arxiv.org/abs/2303.09191arxiv:2303.09191Thu, 16 Mar 2023 00:00:00 GMTHuabin Ge, Bobo Hua, Puchun Zhou — In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we introduce a combinatorial geodesic curvature flow in spherical background geometry, which is analogous to the combinatorial Ricci flow of Chow and Luo in [4]. We characterize the sufficient and necessary condition for the convergence of the flow. That is, the prescribed geodesic curvature satisfies certain geometric and combinatorial condition if and only if for any initial data the flow converges exponentially fast to a circle pattern with given total geodesic curvature on each circle. Our result could be regarded as a resolution of the problem in the spherical case. As far as we know, this is the first combinatorial curvature flow in spherical background geometry with fine properties, and it provides an algorithm to find the desired ideal circle pattern.Ricci flowOther curvature flowsLong-time existence & convergenceDiscrete & graph Ricci