70net永乐高2024年第 3 期数理大讲堂

报告题目：On non-elliptically symplectic manifolds

报告人： 王宏玉教授 （扬州大学）

主持人： 丁青教授 （70net永乐高 ）

讲座时间：2024年1月15日 14：30，

讲座地点 3B205

摘要：Let M be a closed symplectic manifold of dimension 2n with non-ellipticity. We can define an almost K¨ahler structure on M by using the given symplectic form. Using Darboux coordinate charts, we deform the given almost K¨ahler structure to obtain a homotopy equivalent Lipschitz K¨ahler structure on the universal covering of M. Analogous to Teleman’s L 2 -Hodge decomposition on PL manifolds or Lipschitz Riemannian manifolds, we give a L 2 -Hodge decomposition theorem on the universal covering of M w.r.t. the measurable K¨ahler metric. Using an argument of Gromov, we give a vanishing theorem for L 2 harmonic p-forms, p 6= n (resp. a non-vanishing theorem for L 2 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.