A rigidity theorem for Einstein 4-manifolds with sectional curvature of a fixed sign, and its consequences

2025/03/12 by Luca F. Di Cerbo, Di Cerbo, Luca F.
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #math.GT

paper · pdf · doi:10.48550/arxiv.2503.09570

openalex publication_date 2025/03/12 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

Any oriented 4-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality (|s|)/(√(6))≥|W⁺|+|W^-|, where s, W⁺ and W^- are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed 4-dimensional Einstein metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein 4-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality.

Related

• Four-dimensional Einstein manifolds with sectional curvature bounded from above
• Einstein Manifolds, Self-Dual Weyl Curvature, and Conformally Kaehler Geometry
• Kählerity of Einstein four-manifolds
• Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry
• Weyl Curvature, Einstein Metrics, and Seiberg-Witten Theory