Manifolds with 1/4-pinched curvature are space forms

2008/07/17 by Simon Brendle, Richard Schoen · 14 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.1090/s0894-0347-08-00613-9

openalex publication_date 2008/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma g 0 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M,g₀)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact Riemannian manifold with pointwise <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash 4"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1/4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -pinched sectional curvatures. We show that the Ricci flow deforms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g 0"> <mml:semantics> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">g₀</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Böhm and Wilking.

Citations

• Ricci flow with surgery on three-manifolds

Cited by

• Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow
• Compression bounds for Lipschitz maps from the Heisenberg group to L₁
• Complete quasi-Yamabe gradient solitons with bounded scalar curvature
• Non-negative Ricci curvature on closed manifolds under Ricci flow
• The local entropy along Ricci flow---Part B: the pseudo-locality theorems
• Developments around positive sectional curvature
• Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere
• On the blow-up of four-dimensional Ricci flow singularities
• A general convergence result for the Ricci flow in higher dimensions
• Complex Manifolds With Negative Curvature Operator
• Classification of manifolds with weakly 1/4-pinched curvatures
• Lie-algebraic curvature conditions preserved by the Hermitian curvature flow
• The Weyl tensor of gradient Ricci solitons
• Ricci flow [wikipedia]
• Simon Brendle [wikipedia]

Related

• Manifolds with 1/4-pinched Curvature are Space Forms
• On some new Ricci flow invariant curvature conditions
• Manifolds with positive curvature operators are space forms
• Manifolds with positive curvature operators are space forms
• Is a complete Riemannian manifold with positively pinched Ricci curvature compact