On Chern-Yamabe problem

TL;DR

This paper explores an analogue of the Yamabe problem for complex manifolds, establishing existence results for metrics with constant Chern scalar curvature in non-positive cases and discussing challenges in positive curvature scenarios.

Contribution

It introduces the Chern-Yamabe problem for complex manifolds and proves existence of solutions when the expected scalar curvature is non-positive.

Findings

Existence of constant Chern scalar curvature metrics when the curvature is non-positive.

The problem includes cases where the Kodaira dimension is non-negative.

Partial results and remarks on the positive curvature case, including non-uniqueness issues.

Abstract

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when the expected constant Chern scalar curvature is non-positive. In particular, this includes the case when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.