Some new results on domains in complex space with non-compact automorphism group. (English) Zbl 1035.32019

Kim, Kang-Tae; Krantz, Steven G.

J. Math. Anal. Appl. 281, No. 2, 417-424 (2003).

Let \(\Omega\subset\mathbb C^n\) be a bounded domain which is strictly pseudoconvex in a neighborhood of a given point \(p\in\partial \Omega \). Let \(G\) be the group of biholomorphic automorphisms of \(\Omega\) and assume that \(p\) is an orbit accumulation boundary point, i.e. \(p\) is contained in the closure of some \(G\)-orbit in \(\Omega \). By a theorem of B. Wong [Invent. Math. 41, 253–257 (1977; Zbl 0385.32016)] and J.-P. Rosay [Ann. Inst. Fourier 29, 91–97 (1979; Zbl 0402.32001)], \(\Omega \) and \(\mathbb B\) are biholomorphically equivalent.
The authors give a new proof of this fact, based on the theorem of L. Bers that two pseudoconvex domains in \(\mathbb C^n\) are biholomorphically equivalent if their \(\mathbb C\)-algebras of holomorphic functions are isomorphic. In the above situation \(\Omega\) is globally pseudoconvex [see R. E. Greene and S. G. Krantz, Editoria Elettronica, Semin. Conf. 8, 108–135 (1991; Zbl 0997.32012)].
The method of establishing an isomorphism between function algebras can also be applied in the following situation: Without the above boundary condition on \(\Omega\) let \(p'\) be a smooth boundary point of a bounded symmetric domain \(E\) and \(X\) and \(Y\) the maximal analytic varieties in \(\partial \Omega\) and \(\partial E\) passing through \(p\) and \(p'\) respectively. If \(U\cap\overline\Omega\) and \(U'\cap\overline E\) are biholomorphically equivalent for suitable neighborhoods \(U\) of \(X\) and \(U'\) of Y, then \(\Omega\) and \(E\) are biholomorphically equivalent.
It is also shown that every hyperbolic orbit boundary point of a bounded domain in \(\mathbb C^n\) with finite boundary type is a peak point, and that under some additional assumptions every parabolic orbit accumulation boundary point of a pseudoconvex domain in \(\mathbb C^n\) with \(C^\infty\)-boundary is of finite type.

Reviewer: Eberhard Oeljeklaus (Bremen)

MSC:

32T05	Domains of holomorphy
32A38	Algebras of holomorphic functions of several complex variables
32M15	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32T15	Strongly pseudoconvex domains
32T25	Finite-type domains

Keywords:

boundary orbit accumulation point; peak function; boundary point of finite type

Citations:

Zbl 0385.32016; Zbl 0402.32001; Zbl 0997.32012

References:

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