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arXiv · 2505.24762

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

Wenjun Li·Rongyuan Liu·Guohao Chen·Aijin Lin
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Abstract

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.

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Wenjun Li, Rongyuan Liu, Guohao Chen, Aijin Lin. 2025-07-05. Branched $α$-combinatorial Ricci flows on closed surfaces with Euler characteristic $χ\le 0$. https://arxiv.org/abs/2505.24762

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