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We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.

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 $f:\C\> X$.