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

TL;DR

This paper introduces branched alpha-flows on closed surfaces with non-positive Euler characteristic, proving their long-term existence, convergence, and applications to prescribed curvature problems, extending previous alpha-flow results.

Contribution

It generalizes previous alpha-flow results by introducing branched flows and establishes their long-term behavior and applications to prescribed curvature on surfaces with \\chi \\le 0.

Findings

Proved long-time existence and convergence of branched alpha-flows.

Established admissibility conditions for prescribed curvatures.

Demonstrated exponential convergence to target metrics.

Abstract

In this paper we introduce the branched $\alpha$-flows on closed surfaces with Euler characteristic \(\chi \leq 0\). 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.