On non-elliptic symplectic manifolds
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. Hence, we have a $\G=π_1(M)$-invariant almost Kähler structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate charts, we globally deform the given almost Kähler structure on $\ti M$ off a Lebesgue measure zero subset to obtain a $\G$-invariant Lipschitz Kähler flat structure on $\ti M$ which is $\G$-homotopy equivalent to the given almost Kähler structure. Analogous to Teleman's $L^2$-Hodge decomposition on PL manifolds or Lipschitz Riemannian manifolds, we give a $L^2$-Hodge decomposition theorem on $\ti M$ with respect to the Lipschitz Kähler flat metric. Using an argument of Gromov, we give a vanishing theorem for $L^2$ harmonic $p$-forms, $p\not=n$ (resp. a non-vanishing theorem for $L^2$ harmonic $n$-forms) on $\ti M$, then the signed Euler characteristic satisfies $(-1)^nχ(M)\geq0$ (resp. $(-1)^nχ(M)>0$). Similarly, for any closed even dimensional Riemannian manifold $(M, g)$, we can construct a $\G$-invariant Lipschitz Kähler flat structure on the universal covering, $(\ti M, \ti g)$, of $(M, g)$ which is $\G$-homotopy equivalent to and quasi-isometric to the metric $\ti g$. As an application, using Gromov's method 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.