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

In this paper we introduce the branched $α$-flows on closed surfaces with Euler characteristic \(χ≤ 0\). 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 {2015,2015A} on the $α$-flows. In addtion, we study the prescribed curvature problems under the relaxed precondition χ(M)∈ Z via alternative $α$-flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.