https://neckpinch.com/authors/xu-xu/New arXiv papers by Xu Xu, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.25745arxiv:2610.25745Mon, 19 Oct 2026 00:00:00 GMTYuerong Bian, Xu Xu — We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the L^∞ norm.Other curvature flowsKähler geometryStability & uniquenessHarmonic maps & DeTurck trickDiscrete & graph Ricci https://arxiv.org/abs/2607.25691arxiv:2607.25691Tue, 28 Jul 2026 00:00:00 GMTYuerong Bian, Xu Xu — We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an ε-uniformly nondegenerate initial metric, we establish the short-time existence and uniqueness of smooth solutions to the combinatorial Yamabe flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow.Furthermore, under an integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a local well-posedness theory for the hyperbolic combinatorial Yamabe flow on infinitely triangulated surfaces.Yamabe flowStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trickDiscrete & 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/2504.09172arxiv:2504.09172Sat, 12 Apr 2025 00:00:00 GMTZhiwen Xiong, Xu Xu — Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.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/2312.02484arxiv:2312.02484Tue, 05 Dec 2023 00:00:00 GMTXu Xu — Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scalings. We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.Ricci flowYamabe flowOther curvature flowsGeometrizationSurfacesKähler geometryLong-time existence & convergenceDiscrete & 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 https://arxiv.org/abs/2204.08191arxiv:2204.08191Mon, 18 Apr 2022 00:00:00 GMTShengyu Li, Xu Xu, Ze Zhou — This paper studies the combinatorial Yamabe flow on hyperbolic bordered surfaces. We show that the flow exists for all time and converges exponentially fast to conformal factor which produces a hyperbolic surface whose lengths of boundary components are equal to prescribed positive numbers. This provides an algorithm to such problems.Yamabe flowLong-time existence & convergence https://arxiv.org/abs/2110.08474arxiv:2110.08474Sat, 16 Oct 2021 00:00:00 GMTXu Xu — We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.Ricci flowYamabe flowOther curvature flows3-manifoldsSurfacesKähler geometryStability & uniquenessDiscrete & graph Ricci https://arxiv.org/abs/2110.01142arxiv:2110.01142Mon, 04 Oct 2021 00:00:00 GMTYanwen Luo, Xu Xu — Motivated by Luo's combinatorial Yamabe flow on closed surfaces and Guo's combinatorial Yamabe flow on surfaces with boundary, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.Yamabe flowOther curvature flowsSurfacesKähler geometryLong-time existence & convergenceDiscrete & graph Ricci https://arxiv.org/abs/2107.14102arxiv:2107.14102Thu, 29 Jul 2021 00:00:00 GMTTianqi Wu, Xu Xu — Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.Ricci flowYamabe flowOther curvature flowsSingularitiesSurgeryGeometrizationSurfacesKähler geometryLong-time existence & convergenceDiscrete & graph Ricci