Branched $\hat{I}\pm$-combinatorial Ricci flows on closed surfaces with Euler characteristic $\ddot{I}‡\le 0$

Abstract

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