A prescribed curvature flow on hyperbolic surfaces with infinite topological type

TL;DR

This paper introduces a discrete curvature flow on hyperbolic surfaces with infinite topology, proving its well-posedness and convergence to solve the prescribed geodesic curvature problem.

Contribution

It develops a novel prescribed curvature flow adapted to infinite cellular decompositions, enabling construction of hyperbolic metrics on complex surfaces.

Findings

Flow is well-posed under certain conditions

Convergence results established for the flow

Constructs hyperbolic surfaces with prescribed curvature

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.