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Robin functions for complex manifolds and applications

About this Title

Kang-Tae Kim, Department of Mathematics, POSTECH, Pohang, 790-784, Korea, Norman Levenberg, Department of Mathematics, Indiana University, Bloomington, IN 47405 USA and Hiroshi Yamaguchi, Department of Mathematics, Shiga University, Otsu-City, Shiga 520-0862 Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 209, Number 984
ISBNs: 978-0-8218-4965-1 (print); 978-1-4704-0598-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00613-7
Published electronically: July 15, 2010
Keywords: Robin function, variation formula, pseudoconvex, Stein, complex Lie group, complex homogeneous space
MSC: Primary 32U10; Secondary 32M05, 32E10

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Table of Contents

Chapters

• 1. Introduction
• 2. The variation formula
• 3. Subharmonicity of $-\lambda$
• 4. Rigidity
• 5. Complex Lie groups
• 6. Complex homogeneous spaces
• 7. Flag space
• 8. Appendix A
• 9. Appendix B
• 10. Appendix C

Abstract

In a previous Memoirs of the AMS, vol. 92, #448, the last two authors analyzed the second variation of the Robin function $-\lambda (t)$ associated to a smooth variation of domains in ${\mathbb {C}}^n$ for $n\geq 2$. There $\mathcal { D}=\bigcup _{t\in B}(t,D(t))\subset B\times {\mathbb {C}}^n$ was a variation of domains $D(t)$ in ${\mathbb {C}}^n$ each containing a fixed point $z_0$ and with $\partial D(t)$ of class $C^{\infty }$ for $t\in B:=\{t\in {\mathbb {C}}:|t|<\rho \}$. For $z\in \bar {D(t)}$, let $g(t,z)$ be the ${\mathbb {R}}^{2n}$-Green function for the domain $D(t)$ with pole at $z_0$; then \[ \lambda (t):=\lim _{z\to z_0} [g(t,z)-{1\over ||z-z_0||^{2n-2}}].\] In particular, if $\mathcal { D}$ is (strictly) pseudoconvex in $B\times {\mathbb {C}}^n$, it followed that $-\lambda (t)$ is (strictly) subharmonic in $B$. One could then study a Robin function $\Lambda (z)$ associated to a fixed pseudoconvex domain $D\subset {\mathbb {C}}^n$ with $\partial D$ of class $C^{\infty }$ and varying pole $z\in D$. The functions $-\Lambda (z)$ and $\log (-\Lambda (z))$ are real-analytic, strictly plurisubharmonic exhaustion functions for $D$. Part of the motivation and content of our efforts was the study of the Kähler metric $ds^2=\partial \bar \partial \bigl (\log (-\Lambda (z))\bigr )$.

In the current work, we study a generalization of this second variation formula to complex manifolds $M$ equipped with a Hermitian metric $ds^2$ and a smooth, nonnegative function $c$. With this added flexibility, we study pseudoconvex domains $D$ in a complex Lie group $M$ as well as in an $n$-dimensional complex homogeneous space $M$ equipped with a connected complex Lie group $G$ of automorphisms of $M$. We characterize the smoothly bounded, relatively compact pseudoconvex domains $D$ in a complex Lie group which are Stein, and we are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain $D$ in a complex homogeneous space to be Stein. In particular, we describe concretely all the non-Stein pseudoconvex domains $D$ in the complex torus of Grauert; we give a description of all the non-Stein pseudoconvex domains $D$ in the special Hopf manifolds; and we give a description of all the non-Stein pseudoconvex domains $D$ in the complex flag spaces.