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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 $f:\C\> X$.

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This goal of the paper is to show that the automorphisms of the complex of curves in a surface are induced by the self-homeomorphisms of the surface except the surface is the 2-holed torus.

We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions.