\(L_ 2\) cohomology of pseudoconvex domains with complete Kähler metric (Q1345483)

For \(\Omega\) a bounded strictly pseudoconvex domain with smooth boundary in \(\mathbb{C}^n\), equipped with the Bergman metric (which is complete Kaehler), \({\mathcal K}_2^{p,q} (\Omega)\) the space of square integrable harmonic \((p,q)\) forms, the author and Fefferman proved (1983) \[ \dim {\mathcal K}_2^{p,q} (\Omega) = \left\{ \begin{matrix} 0 & \text{ for } & p + q \neq n \\ \infty & \text{ for } & p + q = n \end{matrix} \right\}. \] Here the author presents a more transparent and more elementary proof, using techniques of Gromov. The idea is, for general pseudoconvex domains, to find a sufficient vanishing condition for \(L_2\) cohomology outside the middle degree, in terms of certain extremal problems involving holomorphic functions. This condition is verified for strictly pseudoconvex domains.