The sphere theorems for manifolds with positive scalar curvature

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

2011-02-11

arxiv.org/abs/1102.2424v1

Mathematics

Differential Geometry

35 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

The sphere theorems for manifolds with positive scalar curvature does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The sphere theorems for manifolds with positive scalar curvature, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The sphere theorems for manifolds with positive scalar curvature will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-84307