Integral pinched 3-manifolds are space forms

2007/07/03 by Giovanni Catino, Catino, Giovanni, Zindine Djadli +1
Mathematics · #53C20 #53C21 #53C24 #53C25 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.AP #math.DG #msc:53C20 #msc:53C21 #msc:53C24 #msc:53C25

paper · pdf · doi:10.48550/arxiv.0707.0338

Some misprints in the first version are corrected

openalex publication_date 2007/07/03 · arxiv created 2007/09/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diffeomorphic to a quotient of \Bbb S³.

Citations

• Conformal geometry, contact geometry, and the calculus of variations

Related

• A sphere theorem for three dimensional manifolds with integral pinched curvature
• On 3-manifolds with pointwise pinched nonnegative Ricci curvature
• The sphere theorems for manifolds with positive scalar curvature
• Stability of the Euclidean 3-ball under L2-curvature pinching
• Three-manifolds with non-negatively pinched Ricci curvature