https://neckpinch.com/authors/qing-cui/New arXiv papers by Qing Cui, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1708.09618/arxiv:1708.09618Thu, 31 Aug 2017 00:00:00 GMTQing Cui, Linlin Sun — 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(≥ 4)-dimensional Riemannian manifold M is diffeomorphic to Sⁿ if one of the following conditions holds pointwisely: (i) R₀>(1-2/4(√10-3)n(n-1))Kₘₐₓ; (ii) R/ic^([4])4(n-1)>(1-frac6(√10-3)n-1)Kₘₐₓ. Here Kₘₐₓ, Ric^([k]) and R₀ 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(≥ 4)-dimensional submanifold immersed in M. We prove that M is diffeomorphic to Sⁿ if it satisfies some pinching curvature conditions. The only involved extrinsic quantities in our pinching conditions are the maximal sectional curvature bar Kₘₐₓ and the squared norm of mean curvature vector vert Hvert². More precisely, we show that M is diffeomorphic to Sⁿ if one of the following conditions holds: \beginitemize \item[(1)] R₀≥ (1-2/(n(n-1)))Kₘₐₓ +(n(n-2))/((n-1)²)vert Hvert², and strict inequality is achieved at some point; \item[(2)] R/ic^([2])2≥ (n-2)bar Kₘₐₓ+(n²)/8vert Hvert², and strict inequality is achieved at some point; \item[(3)] R/ic^([2])2 ≥(n(n-3))/(n-2)(bar Kₘₐₓ+vert Hvert²), and strict inequality is achieved at some point. \enditemize 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.Ricci flowCurvature pinchingSphere theoremsScalar curvatureWeak & singular flows