Remarks on the metric induced by the Robin function II

Abstract

Let $D$ be a smoothly bounded pseudoconvex domain in ${\mathbf{C}}^n$, $n>1$. Using the Robin function $\La(p)$ that arises from the Green function $G(z,p)$ for $D$ with pole at $p\in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called $\La$-metric) on $D$. Assume that $D$ is strongly pseudoconvex and $d{s}^2$ denotes the $\La$-metric on $D$. In this article, first we prove that the holomorphic sectional curvature of $d{s}^2$ along normal directions converges to a negative constant near the boundary of $D$. Then, we prove that if $D$ is not simply connected, then any nontrivial homotopy class of ${\pi }_1(D)$ contains a closed geodesic for $d{s}^2$. Finally, we prove that the diminesion of the space of square integrable harmonic $(p,q)$-forms on $D$ relative to $d{s}^2$ is zero except when $p+q=n$ in which case it is infinite.

34 pages