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Guangming Hu

Publications and source records attributed to Guangming Hu.

12 recordsLinked to original sources
arXiv2026

Global rigidity of sphere packings in constant curvature background geometry

In this paper, we study the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of hyperbolic, Euclidean, spherical and ideal hyperbolic sphere packings on a tetrahedron. Moreover, we establish the extended variational principle to obtain the global rigidity of the (ideal) hyperbolic sphere packings on 3-manifolds. We also generalize the conformal tetrahedra to higher dimensions and provide a characterization of conformal $n(\ge3)$-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.

math.GT↗
arXiv2026

Generalized Hyperbolic Conical Circle Packings associated with Finite Polygonal Decompositions of Surfaces with Boundary

Let S be a compact topological surface with finitely many genus and finitely many holes and let D be a polygonal decomposition of S. In this paper, we consider the generalized hyperbolic conical circle packings associated with D. We first show that the boundary value problem has a unique solution k by prescribing total geodesic curvatures of generalized hyperbolic conical circles centered at interior vertices and geodesic curvatures of generalized hyperbolic conical circles centered at boundary vertices. Then we show that such a solution k can be obtained by taking a limit of the packings inductively modified by Thurston's algorithm via an arbitrarily chosen initial generalized hyperbolic conical circle packing associated with D and with given boundary values. Thirdly, we develop the so-called discrete Schwarz-Pick lemma for the solution packing k on D.

math.GT↗
arXiv2025

The asymptoticity of pairs of Teichm\"uller rays

In this paper, we study the limit of Teichm\"uller distance between two points along a pair of Teichm\"uller rays. We obtain an explicit formula for the limiting Teichm\"uller distance when the vertical measured foliations of the quadratic differentials are finite sums of weighted simple closed curves and uniquely ergodic measures. The limit is expressed in terms of ratios of the corresponding moduli and the Teichm\"uller distance between the limit surfaces when the vertical measured foliations are absolutely continuous. Consequently, two Teichm\"uller rays are asymptotic if and only if their vertical measured foliations are modularly equivalent and their limit surfaces coincide, which implies a main result of Masur on the asymptoticity of Teichm\"uller rays determined by uniquely ergodic quadratic differentials. Furthermore, we prove that the infimum of the limiting Teichm\"uller distances can be represented in terms of the distance between the limit surfaces of the Teichm\"uller rays and the detour metric of their endpoints on the Gardiner-Masur boundary, when the initial points of the rays vary along the Teichm\"uller geodesics.

math.CV↗
arXiv2024

Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings

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.

math.DG↗
arXiv2024

Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry

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.

math.DG↗
arXiv2024

The Convergence of Prescribed Combinatorial Ricci Flows for Total Geodesic Curvatures in Spherical Background Geometry

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.

math.GT↗
arXiv2024

Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry

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.

math.DG↗
arXiv2023

Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

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 Verdi\`ere'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.

math.GT↗
arXiv2023

Circle packings and hyperbolic surfaces of finite type

This paper constructs hyperbolic polyhedral metrics via circle packings. We introduce the curvature of circles as a parameter to include all three types of constant curvature curves in the hyperbolic geometry. This provides a unified approach to producing polyhedral metrics for surfaces of broader topological types. The combinatorial total geodesic curvature serves as an effective tool for establishing the existence and uniqueness of the packing.

math.GT↗
arXiv2020

Universal commensurability augmented Teichmüller space and moduli space

It is known that every finitely unbranched covering $α:\widetilde{S}_{g(α)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $Γ_α$ from the Teichmüller space $T(S)$ to the Teichüller space $T(\widetilde{S}_{g(α)})$. Actually, it has been showed that the isometric embedding $Γ_α$ can be extended isometrically to the augmented Teichmüller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichmüller spaces, where the index runs over all finitely unbranched coverings of $S$. Then, we show that the action of the universal commensurability modular group $Mod_{\infty}(S)$ can extend isometrically on $\widehat{T}_{\infty}(S)$. Furthermore, for any $X_{\infty}\in T_{\infty}(S)$, its orbit of the action of the universal commensurability modular group $Mod_{\infty}(S)$ on the universal commensurability augmented Teichmüller space $\widehat{T}_{\infty}(S)$ is dense. Finally, we also construct a directed limit $\widehat{M}_{\infty}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut(π_{1}(S))$ of $Mod_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to produce $\widehat{M}_{\infty}(S)$ as the quotient.

math.GT↗
arXiv2017

Linear differential equations with solutions in weighted Fock spaces

This research is concerned with the nonhomogeneous linear complex differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_{1}f'+A_{0}f=A_{k} $$ in the complex plane. In the higher order case, the mutual relations between coefficients and solutions in weighted Fock spaces are discussed, respectively. In particular, sufficient conditions for the solutions of the second order case $$ f"+Af=0 $$ to be in some weighted Fock space are given by Bergman reproducing kernel and coefficient $A$.

math.CV↗