arXiv subjects

Puchun Zhou

Publications and source records attributed to Puchun Zhou.

13 recordsLinked to original sources

arXiv2026

Infinite Ideal Polyhedra in Hyperbolic 3-Space: Existence and Rigidity

In the seminal work [27], Rivin obtained a complete characterization of finite ideal polyhedra in hyperbolic 3-space by the exterior dihedral angles. Since then,the characterization of infinite hyperbolic polyhedra has become an extremely challenging open problem. By studying ideal circle patterns (ICPs), we characterize the infinite ideal polyhedra (IIP) and resolve this problem. Specifically, we establish the existence and rigidity of embedded ICPs on the plane. We further prove the uniformization theorem for the embedded ICPs, which solves the type problem of infinite ICPs. This is an analog of the uniformization theorem obtained by He and Schramm in [22, 23]. Moreover, we demonstrate that, unlike He-Schramm's work, the type theory for infinite ICPs depends not only on the structure of the cellular decomposition but also on the selection of intersection angles. In fact, we construct Example 4.13 to show the difference. Consequently, we obtain the existence and rigidity of IIP with prescribed exterior angles. Due to the example, our results on the type problem of infinite ICPs and the existence of IIP are sharp. For ICPs with arbitrary angles, our example also demonstrates that the VEL-parabolicity and ICP-parabolicity are not equivalent (while in He and Schramm's settings, VEL-parabolicity and CP-parabolicity are equivalent), indicating that our setting is extremely distinct from He and Schramm's. To prove our results, we develop a uniform Ring Lemma via the technique of pointed Gromov-Hausdorff convergence for ICPs.

math.GT↗

arXiv2026

Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space

Since Thurston pioneered the connection between circle packing (abbr. CP) and three-dimensional geometric topology, the characterization of CPs and hyperbolic polyhedra has become increasingly profound. Some milestones have been achieved, for example, Rodin-Sullivan \cite{Rodin-Sullivan} and Schramm \cite{schramm91} proved the rigidity of infinite CPs with the intersection angle $Θ=0$. Rivin-Hodgson \cite{RH93} fully characterized the existence and rigidity of compact convex polyhedra in $\mathbb{H}^3$. He \cite{He} proved the rigidity and uniformization theorem for infinite CPs with $0\leqΘ\leq π/2$. Therefore, the remaining unresolved issues are the rigidity and uniformization theorems for infinite CPs with $0\leqΘ<π$, as well as for infinite hyperbolic polyhedra. In fact, He specifically claimed in the abstract of \cite{He} that ``in a future paper, the techniques of this paper will be extended to the case when $0\leqΘ<π$. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space". The objective of the article is to accomplish the work claimed in \cite{He} by proving the rigidity and uniformization theorem for infinite CPs with $0\leqΘ<π$, as well as infinite trivalent hyperbolic polyhedra. We will pay special attention to CPs whose contact graphs are disk triangulation graphs. Such CPs are called regular because they exclude some singular configurations and correspond well to hyperbolic polyhedra. We will establish the existence and rigidity of infinite regular CPs. Moreover, we will prove a uniformization theorem for regular CPs, which solves the classification problem for regular CPs. Thereby, the existence and rigidity of infinite convex trivalent polyhedra are obtained.

math.GT↗

arXiv2025

Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

In his seminal work \cite{Ri96}, 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.

math.GT↗

arXiv2025

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

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.

math.GT↗

arXiv2025

Combinatorial Ricci flows on infinite disk triangulations

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.

math.GT↗

arXiv2024

Asymptotic behavior of discrete Schrödinger equations on the hexagonal triangulation

In this article, we prove the decay estimate for the discrete Schrödinger equation (DS) on the hexagonal triangulation. The $l^1\rightarrow l^\infty$ dispersive decay rate is $\left\langle t\right\rangle^{-\frac{3}{4}}$, which is faster than the decay rate of DS on the 2-dimensional lattice $\mathbb{Z}^2$, which is $\left\langle t\right\rangle^{-\frac{2}{3}}$, see [32]. The proof relies on the detailed analysis of singularities of the corresponding phase function and the theory of uniform estimates on oscillatory integrals developed by Karpushkin [15]. Moreover, we prove the Strichartz estimate and give an application to the discrete nonlinear Schrödinger equation (DNLS) on the hexagonal triangulation.

math.AP↗

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↗

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 rigidity of Doyle circle packings on the infinite hexagonal triangulation

Peter Doyle conjectured that locally univalent circle packings on the hexagonal lattice only consist of regular hexagonal packings and Doyle spirals, which is called the Doyle conjecture. In this paper, we prove a rigidity theorem for Doyle spirals in the class of infinite circle packings on the hexagonal lattice whose radii ratios of adjacent circles have a uniform bound. This gives a partial answer to the Doyle conjecture. Based on a new observation that the logarithmic of the radii ratio of adjacent circles is a weighted discrete harmonic function, we prove the result via the Liouville theorem of discrete harmonic functions.

math.GT↗

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↗

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 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.

math.GT↗

arXiv2023

Combinatorial Calabi flows for ideal circle patterns in spherical background geometry

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.

math.GT↗

arXiv2023

A combinatorial curvature flow in spherical background geometry

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.

math.GT↗