https://neckpinch.com/authors/linlin-sun/New arXiv papers by Linlin Sun, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2105.09610/arxiv:2105.09610Thu, 20 May 2021 00:00:00 GMTYong Luo, Linlin Sun, Jiabin Yin — As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo proved that n-dimensional compact self-shrinkers in ℝⁿ⁺¹ with scalar curvature bounded from above or below by some constant are isometric to the round sphere Sⁿ(√n), which implies that n-dimensional compact self-shrinkers in ℝⁿ⁺¹ with constant scalar curvature are isometric to the round sphere Sⁿ(√n)(see also). Complete classifications of n-dimensional translating solitons in ℝⁿ⁺¹ with nonnegative constant scalar curvature and of n-dimensional self-expanders in ℝⁿ⁺¹ with nonnegative constant scalar curvature were given by Martín, Savas-Halilaj and Smoczyk and Ancari and Cheng, respectively. In this paper we give complete classifications of n-dimensional complete self-shrinkers in ℝⁿ⁺¹ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Martín, Savas-Halilaj and Smoczyk and Ancari and Cheng.Mean curvature flowSelf-shrinkers & translatorsSingularitiesScalar curvature https://neckpinch.com/papers/1902.00645/arxiv:1902.00645Sat, 02 Feb 2019 00:00:00 GMTHongbing Qiu, Linlin Sun — In this paper, we firstly prove that every hyper-Lagrangian submanifold L²ⁿ (n > 1) in a hyperkähler 4n-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the complex phase map of self-shrinking surfaces in ℝ⁴. Last but not least, by using the previous rigidity result, we show that the mean curvature flow from a closed surface with the image of the complex phase map contained in S²∖S¹₊ in a hyperkähler 4-manifold does not develop any Type \Rmn1 singularity.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–YauSelf-shrinkers & translatorsSingularitiesKähler geometryStability & uniqueness 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