Infinite circle packings on surfaces with conical singularities

Philip L. Bowers; Lorenzo Ruffoni

doi:10.1016/j.comgeo.2024.102160

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arXiv · 2305.03505

Infinite circle packings on surfaces with conical singularities

Philip L. Bowers·Lorenzo Ruffoni

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Abstract

We show that given an infinite triangulation $K$ of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than $\pi$ at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of $K$. Moreover, if $K$ is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation $K$ and a conformal class $X$ such that there are infinitely many conical hyperbolic structures in the conformal class $X$ with a circle packing in the combinatorics of $K$. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.

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BibTeX RIS

Philip L. Bowers, Lorenzo Ruffoni. 2023-05-05. Infinite circle packings on surfaces with conical singularities. https://doi.org/10.1016/j.comgeo.2024.102160

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