The sphere theorems for manifolds with positive scalar curvature

2011/02/11 by Juan-Ru Gu, Gu, Juan-Ru, Hongwei Xu +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.1102.2424

Abstract

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if Mⁿ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R₀>σ_nK_max, where σ_n∈ ((1)/(4),1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if Mⁿ(n≥3) is a compact manifold whose (n-2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric^(n-2)_min>τ_n(n-2)R₀, where τ_n∈ ((1)/(4),1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors.

Cited by

• Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem

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