Journal of Differential Geometry

The sphere theorems for manifolds with positive scalar curvature

Juan-Ru Gu and Hong-Wei Xu

Full-text: Open access

Abstract

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0 \gt \sigma_n K_{\rm max}$, where $\sigma_n \in (\frac{1}{4} , 1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We also provide a partial answer to Yau’s conjecture on the pinching theorem. Moreover, we prove that if $M^n(n \ge 3)$ is a compact manifold whose $(n − 2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n−2)}_{\rm min} \gt \tau_n(n −2)R_0$, where $\tau_n \in(\frac{1}{4} , 1)$ 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.

Article information

Source
J. Differential Geom., Volume 92, Number 3 (2012), 507-545.

Dates
First available in Project Euclid: 28 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1354110198

Digital Object Identifier
doi:10.4310/jdg/1354110198

Mathematical Reviews number (MathSciNet)
MR3005061

Zentralblatt MATH identifier
1315.53029

Citation

Gu, Juan-Ru; Xu, Hong-Wei. The sphere theorems for manifolds with positive scalar curvature. J. Differential Geom. 92 (2012), no. 3, 507--545. doi:10.4310/jdg/1354110198. https://projecteuclid.org/euclid.jdg/1354110198


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