Combinatorial p-th Calabi flows on surfaces

Abstract

For triangulated surfaces and any p > 1, we introduce the combinatorial p-th Calabi flow which precisely equals 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 = 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.

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

“…Therefore, lim k→∞ u(t n k ) = u. We use Lin-Zhang's trick in [28] to prove lim t→∞ u(t) = u. Suppose otherwise, there exists δ > 0 and ξ n → +∞ such that |u(ξ n ) − u * | > δ.…”

“…To find effective algorithms searching for polyhedral metrics with prescribed combinatorial curvatures on surfaces, Chow-Luo [6] introduced the combinatorial Ricci flow. Motivated by Chow-Luo's original work [6], Ge [9,10] and Ge-Xu [16] introduced the combinatorial Calabi flow, Lin-Zhang [28] introduced the combinatorial p-th Calabi flow and Wu-Xu [36] introduced the fractional combinatorial Calabi flow on surfaces. These combinatorial curvature flows were initially introduced for Thurston's circle packings.…”

Combinatorial Ricci flows for ideal circle patterns

“…background geometry. After that, so-called combinatorial p-th Calabi flows are also studied in [23,13], which generalize the work in [15].…”

Fractional combinatorial Calabi flow on surfaces

“…This implies that the fractional combinatorial Calabi flow (1.10) is a nonlocal combinatorial curvature flow in general. The fractional combinatorial Calabi flow (1.10) is different from the combinatorial p-th Calabi flow defined by discrete p-Laplace operator in [7,28]. Motivated by Definition 2, we further introduce a fractional combinatorial Calabi flow for decorated and hyper-ideal hyperbolic polyhedral metrics on 3-dimensional manifolds in [42], where the basic properties of the flow are also established.…”