Circle packing and Teichmüller space

Abstract

We are interested in a circle packing on a projective Riemann surface, which will be a particular configuration of circles such that all complementary regions are curvilinear triangles. It enjoys both rigid and flexible properties in connection with Teichmüller spaces. The main purpose of this chapter is to discuss such interesting properties from geometric viewpoints. The organization is as follows. After reviewing the basics we discuss rigidity results together with density on constant curvature surfaces due to Koebe, Andreev, Thurston and Brooks. We then discuss flexibility by constructing moduli spaces and formulate a conjecture along with some supporting evidence.