The Ricci flow with metric torsion on closed surfaces

TL;DR

This paper extends the uniformization theorem to surfaces with metric torsion using a modified Ricci flow, showing that such surfaces admit metrics of constant curvature under certain conditions.

Contribution

It introduces an adapted Ricci flow approach to prove uniformization for surfaces with metric torsion, generalizing classical results.

Findings

Existence of constant curvature metrics with metric torsion on closed surfaces

Extension of uniformization theorem to non-positive Euler characteristic cases

Development of a specialized Ricci flow method for torsion-including connections

Abstract

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an adapted form of the Ricci flow.