Circle Pattern Theorem for Quasi-simplicial Triangulated Surfaces

TL;DR

This paper extends the Circle Pattern Theorem to quasi-simplicial triangulations, allowing for loops and multiple edges, by using Delta complexes and covering techniques to characterize curvature conditions.

Contribution

It generalizes the theorem to a broader class of triangulations, removing previous restrictions and employing new methods for analysis.

Findings

Curvature image characterized by KAT inequalities on all vertex subsets

Extension of the theorem to quasi-simplicial triangulations with loops and multiple edges

Use of Delta complexes and finite covering techniques for proof

Abstract

The Circle Pattern Theorem characterizes the existence and rigidity of circle patterns with prescribed intersection angles on simplicial triangulations of closed surfaces. In this paper we extend the theorem to quasi-simplicial triangulations -- triangulations that may contain loops and multiple edges, but whose lifts to the universal cover are simplicial. Chow and Luo first considered such triangulations -- under the name \emph{generalized triangulations} (J.~Differential Geom.~\textbf{63}(1):97--129, 2003) -- but with the strong restriction that any three vertices determine at most one triangle; this condition keeps the combinatorics within the simplicial complex framework and consequently excludes most quasi-simplicial triangulations. We remove this restriction, work instead with the more flexible framework of Delta complexes, and use a finite covering technique to reduce the problem to the simplicial case. We prove that the curvature image is completely characterized by KAT inequalities imposed on all subsets of the lifted vertex set.