Compact manifolds of dimension n≥ 12 with positive isotropic curvature

2019/09/26 by Hong Huang, Huang, Hong · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1909.12265

openalex publication_date 2019/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the following result: Let (M,g₀) be a compact manifold of dimension n≥ 12 with positive isotropic curvature. Then M is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over \mathbbS¹ or I with generic fiber diffeomorphic to \mathbbS^n-1/Γ such that the total space admits a metric with positive isotropic curvature, where Γ is a finite subgroup of O(n) acting freely on \mathbbS^n-1, and I is the one dimensional closed orbifold with two singular points both with local group ℤ₂ and with |I| a closed interval, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen and a conjecture of Gromov in dimensions n≥ 12. The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities.

Citations

• A Compactness Property for Solutions of the Ricci Flow
• The Geometries of 3-Manifolds

Cited by

• On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
• Positive curvature conditions on contractible manifolds

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