https://neckpinch.com/authors/chao-zheng/New arXiv papers by Chao Zheng, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2604.02632arxiv:2604.02632Fri, 03 Apr 2026 00:00:00 GMTYi Li, Jie Wang, Pingsan Yuan, Chao Zheng — In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent characterization of the problem, namely, the solution to the Calabi flow exists globally in time and converges if and only if there exists a weight function that realizes the prescribed curvature. In particular, for constant curvature weights, we prove that the solution to the Calabi flow exists globally in time and converges under certain topological conditions.Other curvature flowsKähler geometryKähler–Einstein & K-stabilityDiscrete & graph Ricci https://arxiv.org/abs/2507.18495arxiv:2507.18495Thu, 24 Jul 2025 00:00:00 GMTXu Xu, Chao Zheng — The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.Ricci flowOther curvature flowsSurfacesKähler geometryStability & uniquenessLong-time existence & convergenceDiscrete & graph Ricci https://arxiv.org/abs/2401.05056arxiv:2401.05056Wed, 10 Jan 2024 00:00:00 GMTXu Xu, Chao Zheng — In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial α-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvatures, we introduce the combinatorial α-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial α-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial α-Ricci flow with surgery. As an application of the combinatorial α-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvatures. We further introduce the combinatorial α-Calabi flow with surgery and study its longtime behavior.Ricci flowOther curvature flowsSingularitiesSurgerySurfacesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability https://arxiv.org/abs/2309.08901arxiv:2309.08901Sat, 16 Sep 2023 00:00:00 GMTTe Ba, Chao Zheng — Generalized circle packings were introduced in as a generalization of tangential circle packings in hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow, fractional combinatorial Calabi flow and combinatorial p-th Calabi flow for generalized hyperbolic circle packings. We establish several equivalent conditions regarding the longtime behaviors of these flows. This provides effective algorithms for finding the generalized circle packings with prescribed total geodesic curvatures.Other curvature flowsKähler geometryLong-time existence & convergenceKähler–Einstein & K-stabilityDiscrete & graph Ricci https://arxiv.org/abs/2309.01205arxiv:2309.01205Sun, 03 Sep 2023 00:00:00 GMTXu Xu, Chao Zheng — Motivated by Guo-Luo's generalized circle packings on surfaces with boundary, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.Ricci flowOther curvature flows3-manifoldsSurfacesKähler geometryScalar curvatureStability & uniquenessBoundaries & non-compactLong-time existence & convergenceDiscrete & graph Ricci https://arxiv.org/abs/2308.02271arxiv:2308.02271Fri, 04 Aug 2023 00:00:00 GMTXu Xu, Chao Zheng — Inversive distance circle packings introduced by Bowers-Stephenson are natural generalizations of Thurston's circle packings on surfaces. To find piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial p-th Calabi flow for the Euclidean inversive distance circle packings. Due to the singularities possibly developed by these combinatorial curvature flows, the longtime existence and convergence of these combinatorial curvature flows have been a difficult problem for a long time. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Using the discrete conformal theory recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics on surfaces, we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. This provides effective algorithms for finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.Other curvature flowsSingularitiesSurgeryGeometrizationSurfacesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stabilityDiscrete & graph Ricci https://arxiv.org/abs/2208.03643arxiv:2208.03643Sun, 07 Aug 2022 00:00:00 GMTXu Xu, Chao Zheng — In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the (−1,−1,−1) type generalized circle packing metric introduced by Guo-Luo. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.Ricci flowOther curvature flowsSurfacesKähler geometryLong-time existence & convergenceDiscrete & graph Ricci