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 K\"{a}hler metric (the so-called $\La$-metric) on $D$. Assume that $D$ is strongly pseudoconvex and $ds^2$ denotes the $\La$-metric on $D$. In this article, first we prove that the holomorphic sectional curvature of $ds^2$ 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 $ds^2$. Finally, we prove that the diminesion of the space of square integrable harmonic $(p, q)$-forms on $D$ relative to $ds^2$ is zero except when $p+q=n$ in which case it is infinite.