In this paper, we study the symplectic cohomologies and symplectic harmonic forms which introduced by Tseng and Yau. Based on this, we get if (M 2n , ω) is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfies the inequality (−1) n χ(M 2n ) ≥ 0. (2000): 53D05. AMS Classification

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

In this paper, we define the generalized Lejmi's P J operator on a compact almost Kähler 2n-manifold. We get that J is C ∞ -pure and full if dim ker P J = b 2 − 1. Additionally, we investigate the relationship between J-anti-invariant cohomology introduced by T.-J. Li and W. Zhang and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic 4-manifold. (2000): 53C55, 53C22. AMS Classification

In this paper, we calculate the dimension of the J-anti-invariant cohomology subgroup H − J on T 4 . Inspired by the concrete example, T 4 , we get that: On a closed symplectic 4-dimensional manifold (M, ω), h − J = 0 for generic ω-compatible almost complex structures. (2000): 53C55, 53D05. AMS ClassificationKeywords: almost Kähler four-manifold, deformations of almost complex structures, dimension of J-anti-invariant cohomology.

Let M n be a complete and noncompact hyper-surface immersed in R n+1 . We should show that if M is of finite total curvature and Ricci flat, then M turns out to be a hyperplane. Meanwhile, the hyper-surfaces with the vanishing scalar curvature is also considered in this paper. It can be shown that if the total curvature is sufficiently small, then by refined Kato's inequality, conformal flatness and flatness are equivalent in some sense. And those results should be compared with Hartman and Nirenberg's similar results with flat curvature assumption.