In this paper, we show that any compact K\"ahler
manifold homotopic to a compact Riemannian manifold with negative
sectional curvature admits a K\"ahler-Einstein metric of general
type. Moreover, we prove that, on a compact symplectic manifold $X$
homotopic to a compact Riemannian manifold with negative sectional
curvature, for any almost complex structure $J$ compatible with the
symplectic form, there is no non-constant $J$-holomorphic entire curve $f:\C\> X$.