Weighted Combinatorial Calabi Flow on Surfaces

Abstract

Motivated by the work of Chow and Luo [J. Differential Geom., 2003, 63(1): 97–129], Ge and his collaborators ([Trans. Amer. Math. Soc., 2018, 370(2): 1377–1391], [Differential Geom. Appl., 2016, 47: 86–98], [Adv. Math., 2018, 333: 523–538]) introduced the combinatorial Calabi flow to study circle patterns in Euclidean and hyperbolic background geometries. Recently, Popelensky [Filomat, 2023, 37(25): 8675–8681] further developed a weighted combinatorial Ricci flow. Inspired by these contributions, we define a weighted combinatorial Calabi flow to investigate circle patterns. This paper addresses two cases: Euclidean and hyperbolic background geometries. In Euclidean background geometry, we prove that the flow exists for all time, and the flow converges if and only if a constant curvature circle pattern metric exists. Moreover, we establish that the prescribed flow converges if and only if the prescribed curvature is attainable. In hyperbolic background geometry, we prove that the flow exists for all time, and the flow converges if and only if a zero curvature circle pattern metric exists. Additionally, in hyperbolic background geometry, for a prescribed curvature, we show that the prescribed flow converges if and only if the curvature is attainable.

Keywords

Combinatorial Calabi flow / weighted flow / circle pattern / combinatorial curvature flow

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Peking University