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

This paper investigates branched $α$-flows on branched weighted triangulated closed surfaces with Euler characteristic \(χ\leq 0\), focusing on establishing their connections with topological-combinatorial structures and geometric structures. In Euclidean (\(\mathbb{E}^2\)) and hyperbolic (\(\mathbb{H}^2\)) geometries, we define branched $α$-curvatures and corresponding branched $α$-flows. By introducing branched $α$-potentials, we prove the existence and uniqueness of constant branched $α$-metrics through topological-combinatorial structures, thus avoiding reliance on flow convergence. Key results include: 1. Exponential convergence of branched $α$-flows to constant branched $α$-metrics in both geometries. 2. Strict convexity of branched $α$-potentials, ensuring unique critical points that correspond to constant curvatures. 3. Extension to prescribing 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. These findings bridge discrete circle packing metrics with smooth geometric invariants, providing a unified framework for studying curvature flows on surfaces with branch structures.