Compact K$ä$hler manifolds homotopic to negatively curved Riemannian manifolds

Abstract: In this paper, we show that any compact K$$\"a$$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$$\"a$$hler-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 $f:C \rightarrow X$.