A prescribed curvature flow on hyperbolic surfaces with infinite topological type

2025/05/26 by Zhao, Xinrong, Puchun Zhou, Zhou, Puchun · 2 citations
Mathematics · #51M10 #52C26 #57M50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.20091

openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Infinite combinatorial Ricci flow in spherical background geometry
• Combinatorial Ricci flows on infinite disk triangulations
• Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary
• Circle packings and hyperbolic surfaces of finite type
• Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces
• On circle patterns and spherical conical metrics
• Uniqueness and stability of singular Ricci flows in higher dimensions
• On the uniqueness of Ricci flow
• Manifolds with 1/4-pinched Curvature are Space Forms
• Manifolds with positive curvature operators are space forms
• Uniqueness of the Ricci Flow on Complete Noncompact Manifolds
• Combinatorial Yamabe Flow on Surfaces
• Combinatorial Ricci Flows on Surfaces
• The entropy formula for the Ricci flow and its geometric applications
• Variational principles for circle patterns and Koebe's theorem

Cited by

• Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns
• Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

Related

• Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary
• Combinatorial curvature flows for generalized circle packings on surfaces with boundary
• A Discrete Ricci Flow on Surfaces in Hyperbolic Background Geometry
• Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometry
• Circle packings and hyperbolic surfaces of finite type