Hirzebruch ${\chi }_y$-genus of compact almost Kähler manifold with negative sectional curvature

Let $(X,J,\omega )$ be a closed $2n$-dimensional almost Kähler manifold with negative sectional curvature. We prove that if the Nijenhuis tensor of the almost complex structure is sufficiently small, then the components of the Hirzebruch ${\chi }_y$-genus satisfy the inequality $(-1{)}^{n-p}{\chi }_p(X)\ge 1$ for all $p=0,1,\cdots ,n$. In particular, this result implies the Hopf conjecture in this setting, namely that the Euler number satisfies $(-1{)}^n\chi (X)\ge n+1$. The proof is based on new $L^2$-estimates for harmonic forms on the universal covering, combined with a refined vanishing theorem for the operator $\bar{\mathrm{\partial }}+{\bar{\mathrm{\partial }}}^{\ast }$ and Atiyah’s $L^2$-index theorem. This work extends the classical result of Gromov [J. Differential Geom., 1991] from the Kähler to the almost Kähler setting under the stated smallness condition.