Deformation of discrete conformal structures on surfaces

In Glickenstein (J Differ Geom 87: 201–237, 2011), Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston’s circle packings, Bowers–Stephenson’s inversive distance circle packings and Luo’s vertex scalings as special cases. In this paper, we study the deformation of Glickenstein’s discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein’s discrete conformal structures on triangulated surfaces (Zhang et al. in Graph Models 76: 321–339, 2014) is a generalization of Chow–Luo’s combinatorial Ricci flow for Thurston’s circle packings (Chow and Luo in J Differ Geom 63: 97–129, 2003) and Luo’s combinatorial Yamabe flow for vertex scalings (Luo in Commun Contemp Math 6: 765–780, 2004). We prove that the solution of the combinatorial Ricci flow for Glickenstein’s discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein’s discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures. © 2024, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.