20 papers
Thurston's sphere packing on 3-dimensional manifolds, II
Xiaokai He, Xu Xu, Chao Zheng
In this paper, we study the rigidity of Thurston's hyperbolic sphere packings on 3-dimensional manifolds. We prove that Thurston's hyperbolic sphere packing is locally determined b…
The Calabi flow with prescribed curvature on finite graphs
Yi Li, Jie Wang, Pingsan Yuan +1
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 Cala…
Infinite rigidity of inversive distance circle packings in the Poincaré disk
Yanwen Luo, Xu Xu, Chao Zheng
The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various…
Discrete conformal structures on surfaces with boundary (III) -- Deformation
Xu 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 \cite{X-…
Discrete conformal structures on surfaces with boundary (II) -- Rigidity and Existence
Xu Xu, Chao Zheng
In \cite{X-Z DCS1}, we introduced discrete conformal structures on surfaces with boundary via an axiomatic framework, and provided a classification of such discrete conformal struc…
A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces
Xu 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…
Discrete conformal structures on surfaces with boundary (I) -- Classification
Xu Xu, Chao Zheng
In this paper, we introduce the discrete conformal structures on surfaces with boundary in an axiomatic approach, which ensures that the Poincaré dual of an ideally triangulated su…
A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II
Xu Xu, Chao Zheng
In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of a geodesic disk…
A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces
Xu Xu, Chao Zheng
In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a wei…
Rigidity of generalized Thurston's sphere packings on 3-dimensional manifolds with boundary
Xu Xu, Chao Zheng
Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized Thurston's sphere packings on 3-dimensional manifolds with bou…
Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary
Xu Xu, Chao Zheng
Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then…
Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces
Xu 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 sur…
On the classification of discrete conformal structures on surfaces
Xu Xu, Chao Zheng
Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification.…
Maximal principles in discrete conformal geometry with application to the rigidity of infinite triangulations
Yanwen Luo, Xu Xu, Chao Zheng
In this paper, maximum principles for Euclidean and hyperbolic discrete conformal structures on polyhedral surfaces are established. These maximum principles unify and generalize t…
Combinatorial curvature flows for generalized circle packings on surfaces with boundary
Xu Xu, Chao Zheng
In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the type generalized circle packi…
A new proof for global rigidity of vertex scaling on polyhedral surfaces
Xu Xu, Chao Zheng
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo, who proved the local rigidity by establishing a variational principle and conjectured…
Prescribing discrete Gaussian curvature on polyhedral surfaces
Xu Xu, Chao Zheng
Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial -curvature…
Density-Based Clustering with Kernel Diffusion
Chao Zheng, Yingjie Chen, Chong Chen +2
Finding a suitable density function is essential for density-based clustering algorithms such as DBSCAN and DPC. A naive density corresponding to the indicator function of a unit $…
Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces
Xu Xu, Chao Zheng
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined o…
Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II
Xu Xu, Chao Zheng
This paper investigates the combinatorial -curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the cl…