Speaker: 王宏玉（扬州大学）

Time: 14:30-16:00, Apr. 8

Venue: 至善楼602

Title: ON NON-ELLIPTIC SYMPLECTIC MANIFOLDS

Abstract:

Let M be a closed symplectic manifold of dimension 2nwith non-ellipticity, We can defne an almost K¤hler structure on Mby using the given symplectic form. Hence, we have a= M)invariant almost Kähler structure on the universal covering, M, of MUsing Darboux coordinate charts, we globally deform the given almostKahler structure on M off a Lebesgue measure zero subset to obtain aT-invariant (measurable) Lipschitz K¤hler fat structure on M which is asingular Kahler structure and [-homotopy equivalent to the given almostKahler structure with Lipschitz condition, restricted to a -invariantopen dense submanifold of M, the K¤hler fat metric is quasi isometricto the given almost Kahler metric. Analogous to Teleman's L?-Hodgedecomposition on PL manifolds or Lipschitz Riemannian manifolds, wegive a L -Hodge decomposition theorem on M with respect to the (mea-surable) Lipschitz Kahler fat metric. As done in Kahler case, using anargument of Gromov, we give a vanishing theorem for ’ harmonic pforms, p ≠n(resp. a non-vanishing theorem for ’ harmonic n-forms)on M,then the signed Euler characteristic satisfes(-1)"x(M)>0(resp.(-1)"x(M)>0). Similarly, for any closed even dimensionalRiemannian manifold (M,g),we can construct a -invariant (measurable) Lipschitz Kahler fat structure on the universal covering,(M, g)of (M, g) which is a singular Kahler structure and T-homotopy equiva-lent to and quasi-isometric to the metric g. As an application, as donein smooth case, using Gromov's method we show that the Chern-Hopfconjecture holds true in closed even dimensional Riemannian manifoldswith nonpositive curvature (resp. strictly negative curvature), it gives apositive answer to a Yau's problem due to S. S. Chern and H. Hopf.