On Chern-Yamabe problem

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.

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• Singular Gauduchon metrics
• On some applications of Gauduchon metrics
• The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds
• Blowing up Chern-Ricci flat balanced metrics