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A prescribed curvature flow on hyperbolic surfaces with infinite topological type

Xinrong Zhao, Puchun Zhou

TL;DR

This work studies the prescribed total geodesic curvature problem for generalized circle packing metrics on hyperbolic surfaces of infinite topological type. It introduces a discrete prescribed curvature flow (PCF) as a noncompact analogue of Ricci flow and proves long-time existence and uniqueness under bounded degree conditions. The authors establish two convergence results: one ensuring convergence to a metric with prescribed curvatures when initial excess is nonnegative, and another under a global upper-bound condition on finite subsets, both yielding generalized circle packings with $T_i=\hat{T}_i$. Consequently, the results provide a method to construct noncompact hyperbolic surfaces with infinite topology and geodesic boundaries or cusps from initial data, extending discrete conformal/curvature flow techniques to infinite settings.

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.

A prescribed curvature flow on hyperbolic surfaces with infinite topological type

TL;DR

This work studies the prescribed total geodesic curvature problem for generalized circle packing metrics on hyperbolic surfaces of infinite topological type. It introduces a discrete prescribed curvature flow (PCF) as a noncompact analogue of Ricci flow and proves long-time existence and uniqueness under bounded degree conditions. The authors establish two convergence results: one ensuring convergence to a metric with prescribed curvatures when initial excess is nonnegative, and another under a global upper-bound condition on finite subsets, both yielding generalized circle packings with . Consequently, the results provide a method to construct noncompact hyperbolic surfaces with infinite topology and geodesic boundaries or cusps from initial data, extending discrete conformal/curvature flow techniques to infinite settings.

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.

Paper Structure

This paper contains 8 sections, 22 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized circle packings on polygons
  4. Generalized Circle Packings on Surfaces
  5. Well-posedness of PCFs
  6. Existence of the prescribed curvature flows
  7. Uniqueness of the PCFs
  8. Convergence of the PCFs

Key Result

Theorem 1.2

Let $\mathcal{D}= (V, E, F)$ be an infinite polygonal cellular decomposition of a noncompact surface $S$. For any initial value $s^0$, there exists a solution $s(t)$, $t\in[0,\infty)$, to the flow eq:PCF. Moreover, if the cellular decomposition has bounded face degree, then the solution to the flow

Figures (4)

  • Figure 1: Circles, horocycles and hypercycles in the hyperbolic space $\mathbb{H}^2$.
  • Figure 2: A generalized circle packing.
  • Figure 3: A generalized pentagon of a generalized circle packing.
  • Figure 4: A generalized circle packing metric on a triangle that all circles are horocycles.

Theorems & Definitions (47)

  • Definition 1.1: Infinite prescribed curvature flow
  • Theorem 1.2: Well-posedness of the PCFs
  • Corollary 1.3
  • Theorem 1.4: Convergence of the hyperbolic PCFs I
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7: Convergence of the hyperbolic PCFs II
  • Definition 1.8
  • Corollary 1.9
  • Definition 1.10
  • ...and 37 more