Combinatorial Calabi flow with surgery on surfaces

Abstract

We study the combinatorial Calabi flow for Euclidean and hyperbolic polyhedral metrics on surfaces, which is an analogue of the smooth surface Calabi flow. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Then we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast to a constant combinatorial curvature metric.Mathematics Subject Classification (2010). 53C44, 52B70.

Discrete Calabi Flow: A Unified Conformal Parameterization Method

“…Ordinary combinatorial Calabi flow may develop singularities. Some triangles degenerate along the flow such that the cotangent value in Equation 3 The long-time existence and convergence of Calabi flow with surgery has been proved in Zhu and Xu's work [ZX18]. In the master's thesis of Tianqi Wu [Wu14], he also proved the finiteness of this edge-flipping surgery in the context of discrete Yamabe flow.…”

“…Then Jin et al proposed discrete Ricci flow on spherical, Euclidean, and hyperbolic backgrounds to deform surfaces conformally in [JKLG08]. In [ Ge18,GH18,ZX18] are equally important. Formally, although the former has a simpler definition by du dt = −K, it has a much more complicated energy u 0 ∑ K i du i compared with the latter.…”

“…Guaranteed convergence. We keep the mesh in Delaunay triangulation using edge-flipping surgery, and the long-time existence and convergence of our discrete Calabi flow is guaranteed [ZX18]. It would neither get trapped in local optimum nor fall into singularities.…”

“…Discrete Calabi flow on piecewise linear (PL) metric [Ge12,ZX18] starts with the definition of the following vertex scaling of lengths:…”

A new class of discrete conformal structures on surfaces with boundary

“…The corresponding long time existence and global convergence of the combinatorial curvature flows has been well-established. See, for instance, [2,9,10,14,28] and others for the combinatorial Ricci (Yamabe) flow and [3][4][5][6]25,34] and others for the combinatorial Calabi flow on closed surfaces. For Guo's vertex scaling of discrete hyperbolic metrics on surfaces with boundary, the long time existence and global convergence of combinatorial Yamabe flow is established in [12,29] and the long time existence and global convergence of combinatorial Calabi flow is established in [17].…”