The sphere theorems for manifolds with positive scalar curvature

TL;DR

This paper establishes new differentiable sphere theorems for manifolds with positive scalar curvature using Ricci flow and stable currents, partially confirming Yau's conjecture and classifying submanifolds with pinched curvatures.

Contribution

It introduces novel curvature pinching conditions involving scalar and Ricci curvatures that guarantee a manifold is diffeomorphic to a spherical space form, extending previous results.

Findings

Manifolds satisfying the scalar curvature and sectional curvature pinching are diffeomorphic to spherical space forms.

New conditions involving the (n-2)-th Ricci curvature ensure diffeomorphism to spherical space forms.

Classification of submanifolds with weakly pinched curvatures, refining existing theorems.

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 $R_0>\sigma_{n}K_{\max}$, where $\sigma_n\in (\frac{1}{4},1)$ 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 $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n-2)}_{\min}>\tau_n(n-2)R_0,$ 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.