报告题目： On non-elliptically symplectic manifolds 

报告人：扬州大学 王宏玉教授

时间：4月18号，下午14：00—15：00

地点：管研楼数学学院1318教室

摘要：Let M be a closed symplectic manifold of dimension 2n with non-ellipticity. We can define an almost Kähler structure on M by using the given symplectic form. Using Darboux coordinate charts, we deform the given almost Kähler structure on the universal covering of M to obtain a Lipschitz Kähler structure on the universal covering of M which is homotopy equivalent to the given almost Kähler structure. Analogous to Teleman's L2-Hodge decomposition on PL manifolds or Lipschitz Riemannian manifolds, we give a L2-Hodge decomposition theorem on the universal covering of M with respect to the Lipschitz Kähler metric. Using an argument of Gromov, we give a vanishing theorem for L2 harmonic p-forms, p≠n (resp. a non-vanishing theorem for L2 harmonic n-forms) on the universal covering of M, then its signed Euler characteristic satisfies (−1)nχ(M)≥0 (resp. (−1)nχ(M)>0).As an application, we show that the Chern-Hopf conjecture holds true in closed even dimensional Riemannian manifolds with nonpositive curvature(resp. strictly negative curvature), it gives a positive answer to a Yau's problem due to S. S. Chern and H. Hopf.

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