Linlin Sun

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 ${\mathbb{R}}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere ${\mathbb{S}}^n(\sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in ${\mathbb{R}}^{n+1}$ with constant scalar curvature are isometric to the round sphere ${\mathbb{S}}^n(\sqrt{n})$(see also). Complete classifications of $n$-dimensional translating solitons in ${\mathbb{R}}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in ${\mathbb{R}}^{n+1}$ 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 ${\mathbb{R}}^{n+1}$ 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.

In this paper, we firstly prove that every hyper-Lagrangian submanifold $L^{2n}(n>1)$ 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 ${\mathbb{R}}^4$. 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 ${\mathbb{S}}^2\setminus {\overline{\mathbb{S}}}_{+}^1$ in a hyperkähler $4$-manifold does not develop any Type \Rmn1 singularity.

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: $$(i)R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};(ii)\frac{Ri{c}^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$ Here $K_{max}$, $Ri{c}^{[k]}$ 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 $|H{|}^2$. More precisely, we show that $M$ is diffeomorphic to $S^n$ if one of the following conditions holds: \beginitemize \item[(1)] $R_0\ge \left(1-\frac{2}{n(n-1)}\right){\bar{K}}_{max}+\frac{n(n-2)}{(n-1{)}^2}|H{|}^2$, and strict inequality is achieved at some point; \item[(2)] $\frac{Ri{c}^{[2]}}{2}\ge (n-2){\bar{K}}_{max}+\frac{n^2}{8}|H{|}^2,$ and strict inequality is achieved at some point; \item[(3)] $\frac{Ri{c}^{[2]}}{2}\ge \frac{n(n-3)}{n-2}\left({\bar{K}}_{max}+|H{|}^2\right),$ 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 $n=4$.