Einstein metrics, harmonic forms, and symplectic four-manifolds

2014/08/31 by Claude LeBrun · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Computer science #Curvature #Differential geometry #Einstein #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematical physics #Mathematics #Moduli space #Omega #Open set #Physics #Pure mathematics #Quantum mechanics #Space (punctuation) #Symplectic geometry #math.DG #math.SG #msc:14J45 #msc:53A30 #msc:53C25 #msc:53C57

paper · pdf · doi:10.1007/s10455-015-9458-0

in Annals of Global Analysis and Geometry (2015)

openalex publication_date 2015/03/10 · arxiv created 2015/04/28 · arxiv updated 2015/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

If M is the underlying smooth oriented 4-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W⁺(ω, ω)> 0, where W⁺ is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic 2-form on (M,h). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on M, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on M.

Citations

• Scalar Curvature and Stability of Toric Varieties
• Some regularity theorems in riemannian geometry
• Spinors and Space-Time
• On Conformally Kaehler, Einstein Manifolds
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I
• Compact Moduli Spaces of Del Pezzo Surfaces and Kähler-Einstein metrics
• Almost-Kahler Anti-Self-Dual Metrics

Cited by

• Cyclic Nielsen realization for del Pezzo surfaces