On Ricci forms of canonical metrics over noncompact complex manifolds

TL;DR

This paper investigates Ricci curvature-related geometric problems on noncompact complex manifolds, including existence of Kähler-Einstein and Hermitian metrics with prescribed Ricci curvature, extending known theorems and conjectures.

Contribution

It advances the understanding of Ricci curvature problems on noncompact complex manifolds, improving existing results and proposing new constructions.

Findings

Established existence results for Kähler-Einstein metrics with negative Ricci curvature

Proved existence of canonical Hermitian metrics with prescribed Ricci curvature

Constructed Hesse-Einstein metrics in affine differential geometry

Abstract

In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature, which can be seen as an improvement of the main theorem in Cheng-Yau [4]; the existence of canonical Hermitian metrics with prescribed Ricci curvature on complete Hermitian manifolds, which can be regarded as noncompact versions of the Gauduchon conjecture on certain complete complex surfaces. Our method can also be used to construct Hesse-Einstein metrics in affine differential geometry.