Remarks on the metric induced by the Robin function II

TL;DR

This paper studies the geometric and topological properties of the $ ext{La}$-metric on strongly pseudoconvex domains, revealing curvature behavior, existence of closed geodesics, and harmonic form dimensions.

Contribution

It proves the boundary behavior of the holomorphic sectional curvature, existence of closed geodesics in non-simply connected domains, and characterizes the space of harmonic forms relative to the $ ext{La}$-metric.

Findings

Holomorphic sectional curvature converges to a negative constant near the boundary.

Non-simply connected domains contain closed geodesics in each nontrivial homotopy class.

Dimension of harmonic $(p,q)$-forms is zero unless $p+q=n$, where it is infinite.

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.