## Journal of Differential Geometry

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

### The sphere theorems for manifolds with positive scalar curvature

#### 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