What are the holomorphic automorphism groups of unit ball, polydisc, and Hartogs domain in $\mathbb{C}^n$ ($n>1$) ? I would be pleased if you tell me.


closed as off-topic by Ben McKay, Alexandre Eremenko, abx, Stefan Waldmann, Pace Nielsen Feb 11 at 17:40

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  • $\begingroup$ Have you tried looking them up, in Rudin's books for example? $\endgroup$ – Ben McKay Feb 10 at 15:13
  • 2
    $\begingroup$ @BenMcKay While that might be one of my first impulses, I am not sure everyone should be expected to know in advance of the right places to look. (That said, this question does seem a bit lacking in context or signs of prior effort) $\endgroup$ – Yemon Choi Feb 11 at 0:20
  • $\begingroup$ This is nearly a duplicate of mathoverflow.net/questions/154612/… $\endgroup$ – Ben McKay Feb 11 at 11:35

An excellent reference for the geometry of the unit ball in $\mathbb{C}^n$ is W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$, Springer, 1970. If $B \subset \mathbb{C}^n$ is the unit ball, and we pick any $a \in B$, let $P_a$ be the orthogonal projection to the orthogonal complement of $a$ in the standard Hermitian metric on $\mathbb{C}^n$. Let $Q_a=I-P_a$. Let $s_a=(1-|a|^2)^{1/2}$. Let $$\phi_a(z)=\frac{a-P_a z - s_a Q_a z}{1-\left<z,a\right>}.$$ Take any $n \times n$ unitary matrix $U$, and finally let $F(z)=U\phi_a(z)$. Then every automorphism of the ball is uniquely expressed as $F$ for some choice of $U$ and $a$, as Rudin proves (in the same notation) on pp. 25-28. Another excellent reference for the geometry in the ball is W. Goldman, Complex Hyperbolic Geometry, Oxford U. Press, 1999.

W. Rudin, Function Theory in Polydiscs, Benjamin, 1969, p. 167 proves that every automorphism of any polydisc $\Delta^n$ (with $\Delta \subset \mathbb{C}$ the open unit disc) is uniquely expressed as the composition of a permutation of the discs and an automorphism of each disk, which is a linear fractional transformation in $SU(1,1)$, as above.

The automorphisms of the Hartogs domains over the classical non-compact Hermitian symmetric spaces are determined in Ahn, Byun and Park, Automorphisms of the Hartogs type domains over classical symmetric domains, Internat. J. Math. 23 (2012), no. 9. They turn out to be the expected classical Lie groups.

  • $\begingroup$ Thank you very much !! $\endgroup$ – artifact Feb 11 at 11:30

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