Existence, Rigidity, and Discrete Schwarz--Pick Lemma for Generalized Hyperbolic Circle Packings with Boundary and Discrete Gaussian Curvatures

Abstract

This paper is concerned with generalized hyperbolic circle packings on compact bordered surfaces, endowed with finite polygonal cellular decompositions. We investigate the problem of realizing generalized hyperbolic circle packings with prescribed geodesic curvatures at boundary vertices, prescribed total geodesic curvatures at interior vertices, and prescribed discrete Gaussian curvatures at the centers of dual circles. We give a necessary and sufficient condition for the existence of such generalized hyperbolic circle packings and show their uniqueness. We also establish a discrete Schwarz--Pick lemma in this setting, including comparison results for vertex curvatures, generalized circle arc lengths, distances, and areas, together with the corresponding rigidity statements.