Combinatorial Calabi flows on surfaces with boundary

Combinatorial curvature flows for generalized circle packings on surfaces with boundary

“…Motivated by the combinatorial Ricci (Yamabe) flow [14,17,32] and the combinatorial Calabi flow [21,32] on surfaces with boundary, we introduce the following combinatorial Definition 1.2. Suppose (Σ, T ) is an ideally triangulated bordered surface with a weight Φ : E → (0, +∞), K ∈ (0, +∞) N is a given function defined on B = {1, 2, ..., N }.…”

“…Motivated by the combinatorial Yamabe flow introduced by Guo [14], Li-Xu-Zhou [17] recently introduced a modified combinatorial Yamabe flow for Guo's vertex scaling on ideally triangulated surfaces with boundary, which generalizes and completes Guo's results [14]. Motivated by Ge [4,5] and Ge-Xu [7], Luo-Xu [21] introduced combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary and proved its global convergence. Motivated by the fractional combinatorial Calabi flow introduced by Wu-Xu [29] for discrete conformal structures on closed surfaces, Luo-Xu [21] further introduced fractional combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary.…”

“…Motivated by Ge [4,5] and Ge-Xu [7], Luo-Xu [21] introduced combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary and proved its global convergence. Motivated by the fractional combinatorial Calabi flow introduced by Wu-Xu [29] for discrete conformal structures on closed surfaces, Luo-Xu [21] further introduced fractional combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary. The global convergence of the fractional combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary is also obtained in [21].…”

“…Motivated by the fractional combinatorial Calabi flow introduced by Wu-Xu [29] for discrete conformal structures on closed surfaces, Luo-Xu [21] further introduced fractional combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary. The global convergence of the fractional combinatorial Calabi flow for Guo's vertex scaling on surfaces with boundary is also obtained in [21]. Recently, Xu [32] introduced a new class of hyperbolic discrete conformal structures on ideally triangulated surfaces with boundary and further introduced the corresponding combinatorial Ricci flow, combinatorial Calabi flow and fractional combinatorial Calabi flow on surfaces with boundary.…”

“…In this paper, motivated by the combinatorial Ricci (Yamabe) flow [14,17,32] and the combinatorial Calabi flow [21,32] on surfaces with boundary, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary, which is the (−1, −1, −1) type generalized circle packing introduced by Guo-Luo [16]. We further prove the longtime existence and global convergence for the solutions of the combinatorial Ricci flow and the combinatorial Calabi flow on ideally triangulated surfaces with boundary.…”