Remarks on the metric induced by the Robin function. II (Q376672)

In [Indiana Univ. Math. J. 60, No. 3, 751--802 (2011; Zbl 1251.32011)] the author and \textit{K. Verma} began to study the properties of the metric induced by the Robin function. In the paper under review, the author presents further properties of this metric as well as the generalizations of the results given in the paper mentioned above.NEWLINENEWLINEFor instance, the author is able to remove the restriction on the convergence to a boundary point along the inner normal when obtaining the boundary behavior of the holomorphic sectional curvature of the metric induced by the Robin function.NEWLINENEWLINEIn the paper it is also shown that on a \(\mathcal C^{\infty}\)-smoothly bounded strongly pseudoconvex domain \(D\) in \(\mathbb C^n\) that is not simply connected, every nontrivial homotopy class in \(\pi_1(D)\) contains a closed geodesic for the metric induced by the Robin function (\textit{G. Herbort} in [Math. Ann. 264, 39--51 (1983; Zbl 0497.32022)] showed the same result for the Bergman metric).NEWLINENEWLINEFinally, using ideas of Donnelly and Gromov, the author presents some result on the dimension of the space of square-integrable harmonic \((p,q)\)-forms relative to the metric induced by the Robin function.