  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. Hence, we have a $\G=\pi_1(M)$-invariant almost K\"ahler
structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate
charts, we globally deform the given almost K\"ahler structure on $\ti M$ off a
Lebesgue measure zero subset to obtain a $\G$-invariant Lipschitz K\"ahler flat
structure on $\ti M$ which is $\G$-homotopy equivalent to the given almost
K\"ahler 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 K\"ahler 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
$(-1)^n\chi(M)\geq0$ (resp. $(-1)^n\chi(M)>0$). Similarly, for any closed even
dimensional Riemannian manifold $(M, g)$, we can construct a $\G$-invariant
Lipschitz K\"ahler 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.
