Branched $α$-combinatorial Ricci flows on closed surfaces with Euler characteristic $χ\le 0$

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