Combinatorial p-th R-curvature Ricci Flows and Calabi Flows on Surfaces

In this paper, we study the convergence of the solutions to combinatorial p-th R-curvature Ricci flows and combinatorial p-th R-curvature Calabi flows on surfaces. R-curvature was introduced by Ge [Int. Math. Res. Not. IMRN, 2017, 2017(11): 3510–3527] which is a modification of the well-known discrete Gaussian curvature on triangulated manifolds. We show that the long time convergence of the solutions to combinatorial p-th R-curvature Ricci flows on surfaces is equivalent to the existence of constant R-curvature metrics. Furthermore, we show that the solutions to combinatorial p-th R-curvature Calabi flows on surfaces in the Euclidean background geometry and hyperbolic background geometry have the long time convergence if and only if there exist constant R-curvature metrics.