Combinatorial \(p\)-th Calabi flows for total geodesic curvatures in hyperbolic background geometry (Q6649693)

In this paper, the authors study what they call generalized circle packings on a hyperbolic surface. The circle packings are said to be ``generalized'' because they include not only usual circles, but also horocycles and hypercycles. The underlying surface has boundary and conical singularities. It has a ``total geodesic curvature'' on each generalized circle of the circle packing and discrete Gaussian curvature at the center of each dual circle. The notion of total geodesic curvature was first introduced in [\textit{X. Nie}, Proc. Am. Math. Soc. 152, No. 2, 843--853 (2024; Zbl 1536.52022)], as a tool for studying the rigidity of circle patterns in a spherical background geometry.\N\NThe authors' purpose is to find circle packings with the above properties. For this, they first prove existence and rigidity for this type of circle packings, using a variational principle. Then, for \(p > 1\), they introduce a combinatorial \(p\)-th Calabi flow in the setting of total geodesic curvature with a hyperbolic background geometry. They use this to find the circle packing with the required properties. The paper contains beautiful ideas.