Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

This paper presents a comprehensive study of the combinatorial $p$-th Calabi flow for both finite and infinite ideal circle patterns. In the finite case, we establish a sharp criterion: the combinatorial $p$-th Calabi flow with $p>1$ converges if and only if a constant curvature metric exists in the underlying geometric background. In the infinite setting, we prove the long-time existence of solutions to the combinatorial $p$-th Calabi flow for $p\ge 2$, representing a significant advance in the theory of curvature flows on infinite structures.