# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Kenzō Adachi, Le problème de Lévi pour les fibrés grassmanniens et les variétés drapeaux, Pacific J. Math. 116 (1985), no. 1, 1-6 (French, with English summary). · Zbl 0513.32021 [2] Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. · Zbl 0196.33801 [3] Bo Berndtsson, Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1633-1662 (English, with English and French summaries). · Zbl 1120.32021 [4] Klas Diederich and Takeo Ohsawa, A Levi problem on two-dimensional complex manifolds, Math. Ann. 261 (1982), no. 2, 255-261. · Zbl 0502.32010 · doi:10.1007/BF01456222 [5] Hans Grauert, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81 (1963), 377-391 (German). · Zbl 0151.09702 [6] André Hirschowitz, Le problème de Lévi pour les espaces homogènes, Bull. Soc. Math. France 103 (1975), no. 2, 191-201. · Zbl 0316.32004 [7] Jae-Cheon Joo, On the Levenberg-Yamaguchi formula for the Robin function, Complex Var. Elliptic Equ. 54 (2009), no. 3-4, 345-353. · Zbl 1163.32302 · doi:10.1080/17476930902759411 [8] Hideaki Kazama, On pseudoconvexity of complex Lie groups, Mem. Fac. Sci. Kyushu Univ. Ser. A 27 (1973), 241-247. · Zbl 0276.32020 [9] H. Kazama, D. K. Kim, and C. Y. Oh, Some remarks on complex Lie groups, Nagoya Math. J. 157 (2000), 47-57. · Zbl 0957.32010 [10] Kang-Tae Kim, Norman Levenberg, and Hiroshi Yamaguchi, Robin functions for complex manifolds and applications, Complex analysis and its applications, OCAMI Stud., vol. 2, Osaka Munic. Univ. Press, Osaka, 2007, pp. 25-42. · Zbl 1160.32022 [11] Norman Levenberg and Hiroshi Yamaguchi, The metric induced by the Robin function, Mem. Amer. Math. Soc. 92 (1991), no. 448, viii+156. · Zbl 0742.31003 · doi:10.1090/memo/0448 [12] Norman Levenberg and Hiroshi Yamaguchi, Robin functions for complex manifolds and applications, Sūrikaisekikenkyūsho Kōkyūroku 1037 (1998), 138-142. CR geometry and isolated singularities (Japanese) (Kyoto, 1996). · Zbl 0953.32018 [13] Daniel Michel, Sur les ouverts pseudo-convexes des espaces homogènes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 10, Aiii, A779-A782 (French, with English summary). · Zbl 0355.32019 [14] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970. · Zbl 0199.40603 [15] Toshio Nishino, Function theory in several complex variables, Translations of Mathematical Monographs, vol. 193, American Mathematical Society, Providence, RI, 2001. Translated from the 1996 Japanese original by Norman Levenberg and Hiroshi Yamaguchi. · Zbl 0972.32001 [16] Takeo Ohsawa, On the Levi-flats in complex tori of dimension two, Publ. Res. Inst. Math. Sci. 42 (2006), no. 2, 361-377. · Zbl 1141.32016 [17] Yum Tong Siu, Pseudoconvexity and the problem of Levi, Bull. Amer. Math. Soc. 84 (1978), no. 4, 481-512. · Zbl 0423.32008 · doi:10.1090/S0002-9904-1978-14483-8 [18] Zibgniew Slodkowski and Giuseppe Tomassini, Minimal kernels of weakly complete spaces, J. Funct. Anal. 210 (2004), no. 1, 125-147. · Zbl 1055.32009 · doi:10.1016/S0022-1236(03)00182-4 [19] Osamu Suzuki, Remarks on examples of $$2$$-dimensional weakly $$1$$-complete manifolds which admit only constant holomorphic functions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1979), no. 3, 253-261. · Zbl 0409.32011 [20] Tetsuo Ueda, Pseudoconvex domains over Grassmann manifolds, J. Math. Kyoto Univ. 20 (1980), no. 2, 391-394. · Zbl 0456.32011 [21] Hiroshi Yamaguchi, Variations of pseudoconvex domains over $${\mathbf C}^n$$, Michigan Math. J. 36 (1989), no. 3, 415-457. · Zbl 0692.31004 · doi:10.1307/mmj/1029004011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.