Complete Kähler manifolds with zero Ricci curvature. I

1990/01/01 by Gang Tian, Shing-Tung Yau · 259 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory #Mathematics #Zero (linguistics) #Ricci curvature #Pure mathematics #Ricci-flat manifold #Sectional curvature #Curvature #Ricci flow #Curvature of Riemannian manifolds #Scalar curvature #Mathematical analysis #Geometry

paper · pdf · doi:10.1090/s0894-0347-1990-1040196-6

published in Journal of the American Mathematical Society 3(3), 579-609 (American Mathematical Society)

openalex publication_date 1990/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

The problem of constructing complete manifolds with zero Ricci curvature is important for both physicists and geometers. When the manifold is compact and Kahler, this problem was solved satisfactorily by the second author in 1976. While it is not difficult to construct explicit examples of noncompact manifolds with zero Ricci curvature, a complete understanding of complete noncompact manifolds with zero Ricci curvature is still needed. . Therefore, immediately after the work in 1976, the second author proposed a scheme to classify these manifolds. This is the first part of the papers being written by the authors on a systematic research on the existence of these metrics. They have natural applications to algebraic geometry which shall be reported on later.

Citations

• Geometry of Manifolds
• Elliptic Partial Differential Equations of Second Order
• On the existence of a complete Kähler metric on non‐compact complex manifolds and the regularity of fefferman's equation
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I

Cited by

• Large Complex Structure Limits of K3 Surfaces
• Perspectives on geometric analysis
• Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities
• A \mathcal C^2,α estimate of the complex Monge-Ampère equation
• A Liouville theorem on asymptotically Calabi spaces
• ALE Ricci-flat Kahler surfaces and weighted projective spaces
• Geometric unification of Higgs bundle vacua
• Compact Manifolds with Unbounded Nilpotent Fundamental Groups and Positive Ricci Curvature
• Shing-Tung Yau [wikipedia]