I take $F$ from $\Omega\subset \mathbb C^n$ to $\mathbb C^n$ to be a holomorphic function such that $$| \det(J_F)|\leq 1,$$ where $J_F$ is the Jacobian matrix of $F$.
My question: Is there any classification of functions of this type?
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4$\begingroup$ What kind of classification do you expect? Up to $\mathrm{Diff}(\Omega)$? There are a lot of such functions: take e.g. any triangular nonlinear pertubation of a triangular linear application $L$ with $|\det(L)|\leq1$. $\endgroup$– Loïc TeyssierCommented Dec 7, 2022 at 10:43
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$\begingroup$ I am interested to understand the type of the functions satisfying this inequality, then any classification will be useful.Thanks for your answer. $\endgroup$– Said KamamCommented Dec 9, 2022 at 12:53
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1 Answer
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This function has constant Jacobian by Liouville. Then it is map of Jacobian 1 composed with a homothety or its differential is degenerate everywhere. The constant Jacobian biholomorphisms are subject of much research, see for example
Rosay, Jean-Pierre, Automorphisms of Cn, a survey of Andersén-Lempert theory and applications. Complex geometric analysis in Pohang (1997), 131–145, Contemp. Math., 222, Amer. Math. Soc., Providence, RI, 1999.
answered Dec 7, 2022 at 17:48
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5$\begingroup$ Verbisty Be careful, $F$ is defined on $\Omega$, not on the whole space $\mathbb{C}^n$. So Liouville theorem does not apply. $\endgroup$ Commented Dec 7, 2022 at 18:12
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$\begingroup$ sorry, i misread the question $\endgroup$ Commented Dec 8, 2022 at 22:05
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$\begingroup$ Misha Verbitsky, thanks a lot, I find the reference that you mentioned very interseting, I didn't know it before. $\endgroup$ Commented Dec 9, 2022 at 12:55