The metric induced by the Robin function.

*(English)*Zbl 0742.31003
Mem. Am. Math. Soc. 448, 156 p. (1991).

The book is devoted to the following surprising relationship between \(\mathbb{R}^{2n}\)-potential theory and pluripotential theory. Let \(D\) be a bounded pseudoconvex domain in \(\mathbb{C}^ n\thickapprox\mathbb{R}^ 2n\) with smooth boundary, \(G(\xi,z)\) be the Green function for \(D\) associated to the Laplace operator, and
\[
\Lambda(\xi){\overset\text{def} =}\lim_{z \to \xi} \left[ G(z,\xi)-{1 \over \| z-\xi\|^{2n-2}}\right]
\]
be the Robin function for \(D\). Then the functions \(-\Lambda(\xi)\) and \(\log(-\Lambda(\xi))\) are strictly plurisubharmonic exhaustion functions in \(D\) [the second author, Mich. Math. J. 36, No. 3, 415-457 (1989; Zbl 0692.31004)]. A new proof of this result is given here which is entirely self-contained. Explicit formulas for the complex Hessians of \(- \Lambda(\xi)\) and \(\log(-\Lambda(\xi))\) are obtained. The following boundary regularity is proved: if \(D=\{\xi\in\mathbb{C}^ n: \psi(\xi)<0\}\), \(|\text{grad }\psi|\neq 0\) for every point of \(\partial D\), then the function \(\Lambda(\xi)[\psi(\xi)]^{2n-2}\) is \(C^ 2\) up to \(\partial D\). To prove these results the technique of variation of domains in \(\mathbb{C}^ n\) is used.

Further the authors study the Kähler metric \[ ds^ 2=\sum_{\alpha,\beta=1}^ n\left[ {\partial^ 2\log(-\Lambda) \over \partial\xi_ \alpha \partial\overline {\xi}_ \beta}\right] d\xi_ \alpha \otimes d\xi_ \beta. \] It is proved that for a wide class of smooth domains (including strictly pseudoconvex and \(\mathbb{R}^{2n}\)-convex ones) this metric is complete. An open problem is completeness of the metric for an arbitrary bounded domain with smooth boundary as well as how this metric is related to biholomorphic invariants of a domain \(D\). Included are explicit calculations for some special examples.

Further the authors study the Kähler metric \[ ds^ 2=\sum_{\alpha,\beta=1}^ n\left[ {\partial^ 2\log(-\Lambda) \over \partial\xi_ \alpha \partial\overline {\xi}_ \beta}\right] d\xi_ \alpha \otimes d\xi_ \beta. \] It is proved that for a wide class of smooth domains (including strictly pseudoconvex and \(\mathbb{R}^{2n}\)-convex ones) this metric is complete. An open problem is completeness of the metric for an arbitrary bounded domain with smooth boundary as well as how this metric is related to biholomorphic invariants of a domain \(D\). Included are explicit calculations for some special examples.

Reviewer: A.Yu.Rashkovsky (Khar’kov)

##### MSC:

31C10 | Pluriharmonic and plurisubharmonic functions |

32T99 | Pseudoconvex domains |

32U05 | Plurisubharmonic functions and generalizations |