Combinatorial p-th Calabi flows on surfaces

For triangulated surfaces and any p > 1, we introduce the combinatorial p-th Calabi flows which precisely equal the combinatorial Calabi flows first introduced in H. Ge's thesis [9] (or see H. Ge [13]) when p = 2. The difficulties for the generalizations come from the nonlinearity of the p-th flow equation when p not equal 2. Adopting different approaches, we show that the solution to the combinatorial p-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H. Ge [13], Ge-Xu [19] and Ge Bua [14] on the combinatorial Calabi flow from p = 2 to any p > 1. (C) 2019 Elsevier Inc. All rights reserved.