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Robin functions for complex manifolds and applications. (English) Zbl 1228.32001
Mem. Am. Math. Soc. 984, vii, 111 p. (2011).
Partly from the authors’ abstract: In [Mem. Am. Math. Soc. 92, No. 448, 156 p. (1991; Zbl 0742.31003)], 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
\[ {\mathcal{D}}=\bigcup_{t\in B}\big(t,D(t)\big)\subset B\times {\mathbb{C}}^n \]
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 \({\mathcal{C}}^{\infty}\) for \(t\in B:=\{t\in {\mathbb{C}}:|t|<\rho\}\). For \(z\in \overline{D(t)}\), let \(g(t,z)\) be the \({\mathbb{R}}^{2n}\)-Green function for the domain \(D(t)\) with pole at \(z_0\); then \[ \lambda (t):=\lim_{z\to z_0}\left[g(t,z)-\frac{1}{\| z-z_0\| ^{2n-2}}\right]. \] 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 \({\mathcal{C}}^{\infty}\) and varying pole \(z\in D\). A surprising result, see already [the third author, Mich. Math. J. 36, No. 3, 415–457 (1989; Zbl 0692.31004)] and also [B. Berndtsson, Ann. Inst. Fourier (Grenoble) 56, No. 6, 1633–1662 (2006; Zbl 1120.32021)] is that the functions \(-\Lambda (z)\) and \(\log (-\Lambda (z))\) are real-analytic, strictly plurisubharmonic exhaustion functions for \(D\). Part of the motivation and content of the authors’ efforts was the study of the Kähler metric \(ds^2=\partial \overline{\partial} \big(\log (-\Lambda (z))\big)\).
Observe that \(\lambda (t)\) is determined by classical Newtonian potential theory in \({\mathbb{R}}^{2n}\). Hence the associated Green function \(g(t,z)\) transforms well under translations of \({\mathbb{R}}^{2n}\), but not under general biholomorphic changes of coordinates in \({\mathbb{C}}^{n}\). Therefore, and in order to be able to live with this handicap, the authors of the current work now study a generalization of the second variation formula to complex manifolds \(M\), equipped with a Hermitian metric \(ds^2\) and a smooth, non-negative function \(c\). With this added flexibility the authors consider 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 \(M\). They are able to characterize the smoothly bounded, relatively compact pseudoconvex domains \(D\) in a complex Lie group which are Stein. They are also able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain \(D\) in a complex homogeneous space to be Stein. In particular, they describe concretely all the non-Stein pseudoconvex domains \(D\) in the complex torus of Grauert. Similarly, they give a description of all the non-Stein pseudoconvex domains \(D\) in the special Hopf manifolds, and a description of all the non-Stein pseudoconvex domains \(D\) in the complex flag spaces.

MSC:
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32U10 Plurisubharmonic exhaustion functions
32M05 Complex Lie groups, group actions on complex spaces
32Q15 Kähler manifolds
32Q28 Stein manifolds
32U05 Plurisubharmonic functions and generalizations
31C99 Generalizations of potential theory
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