Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most $-a^2<0$ carry a nonconstant bounded holomorphic function?

Theorem 1.1: for some finite $m\ge2$ there is a domain $\mathcal T\subset\mathbb C^m$, diffeomorphic to $\mathbb R^{2m}$, with a complete Kahler metric $g$ satisfying $\mathrm{Sec}_g\le -1$ and $H^\infty(\mathcal T)=\mathbb C$: every bounded holomorphic function is constant, so the Caratheodory pseudometric vanishes. This disproves the universal assertion in the one-sided question, and also answers Wu's 1967 bounded-domain question negatively under its hypotheses. The sectional curvatures are unbounded below, so the question under a two-sided bound $-b^2\le\mathrm{Sec}\le -a^2$ remains open; the dimension is large and not made explicit.

Theorem 1.1: for some finite $m\ge2$ there is a domain $\mathcal T\subset\mathbb C^m$, diffeomorphic to $\mathbb R^{2m}$, with a complete Kahler metric $g$ satisfying $\mathrm{Sec}_g\le -1$ and $H^\infty(\mathcal T)=\mathbb C$: every bounded holomorphic function is constant, so the Caratheodory pseudometric vanishes. This disproves the universal assertion in the one-sided question, and also answers Wu's 1967 bounded-domain question negatively under its hypotheses. The sectional curvatures are unbounded below, so the question under a two-sided bound $-b^2\le\mathrm{Sec}\le -a^2$ remains open; the dimension is large and not made explicit.