Branched $\alpha$-combinatorial Ricci flows on closed surfaces with Euler characteristic $\chi \le 0$

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic $\chi \le 0$. Based on the strict convexity of the branched $\alpha$-potentials, we establish the long time existence and convergence of the solutions to the branched $\alpha$-flows, which generalizes Ge and Xu’s main results \cite{2015,2015A} on the $\alpha$-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition $\chi (M)\in \mathbb{Z}$ via alternative $\alpha$-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.