Some sharp differential sphere theorems for nonnegative scalar curvature manifolds

TL;DR

This paper establishes new differential sphere theorems for manifolds with nonnegative scalar curvature, using Ricci flow and pinching conditions, to prove diffeomorphism to spheres in both intrinsic and extrinsic cases.

Contribution

It introduces novel curvature pinching conditions that guarantee a manifold is diffeomorphic to a sphere, extending previous results with optimal constants in certain cases.

Findings

Manifolds satisfying specific scalar curvature pinching are diffeomorphic to spheres.

New intrinsic and extrinsic curvature conditions are sufficient for sphere theorems.

Optimal constants are identified for certain dimensions, notably n=4.

Abstract

In this paper, we obtain several new intrinsic and extrinsic differential sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $n(\ge 4)$-dimensional Riemannian manifold $M$ is diffeomorphic to $S^n$ if one of the following conditions holds pointwisely: $$ (i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$ Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectional curvature, the $k$-th weak Ricci curvature and the normalized scalar curvature. For extrinsic case, i.e., when $M$ is a closed simply connected $n(\ge 4)$-dimensional submanifold immersed in $\bar{M}$. We prove that $M$ is diffeomorphic to $S^n$ if it satisfies some pinching curvature conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature $\bar K_{max}$ and the squared norm of mean curvature vector $\vert H\vert^2$. More precisely, we show that $M$ is diffeomorphic to $S^n$ if one of the following conditions holds: \begin{itemize} \item[(1)] $R_0\ge \left(1-\frac{2}{n(n-1)}\right)\bar{K}_{max} +\frac{n(n-2)}{(n-1)^2}\vert H\vert^2$, and strict inequality is achieved at some point; \item[(2)] $\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar K_{max}+\frac{n^2}{8}\vert H\vert^2,$ and strict inequality is achieved at some point; \item[(3)] $\dfrac{Ric^{[2]}}{2} \ge\frac{n(n-3)}{n-2}\left(\bar K_{max}+\vert H\vert^2\right),$ and strict inequality is achieved at some point. \end{itemize} It is worth pointing out that, in the proof of extrinsic case, we apply suitable complex orthonormal frame and simplify the calculations considerably. We also emphasize that both of the pinching constants in (2) and (3) are optimal for $n=4$.