Wu's bounded-domain question for negatively pinched Kahler manifolds: is a complete simply connected Kahler manifold with pinched negative curvature biholomorphic to a bounded domain?

Theorem 1.1: there is a contractible domain $M\subset\mathbb C^3$ (a Hartogs-type disc bundle over the unit ball of $\mathbb C^2$) with a smooth complete Kahler metric whose real sectional curvatures satisfy $-B\le K\le -A<0$, such that no bounded holomorphic map $M\to\mathbb C^3$ has nowhere-vanishing Jacobian; hence $M$ is not biholomorphic to any bounded domain. This disproves Wu-Yau Conjecture 4.3 and gives a negative answer under the hypotheses of Wu's 1967 question. It does not show absence of bounded holomorphic functions (two base coordinates are bounded), and the example is Stein. Only complex dimension three is constructed.

Theorem 1.1: there is a contractible domain $M\subset\mathbb C^3$ (a Hartogs-type disc bundle over the unit ball of $\mathbb C^2$) with a smooth complete Kahler metric whose real sectional curvatures satisfy $-B\le K\le -A<0$, such that no bounded holomorphic map $M\to\mathbb C^3$ has nowhere-vanishing Jacobian; hence $M$ is not biholomorphic to any bounded domain. This disproves Wu-Yau Conjecture 4.3 and gives a negative answer under the hypotheses of Wu's 1967 question. It does not show absence of bounded holomorphic functions (two base coordinates are bounded), and the example is Stein. Only complex dimension three is constructed.