Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

By P. Ebenfelt, B. Lamel, D. Zaitsev. 2001

83 citations recorded by the source corpus.

AbstractWe show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in $\mathbb{C}$2 at a point p ∈ M are uniquelydetermined by their jets of some finite order at p if and only if M is not Levi-flat near p. This seems to be the first necessary and sufficient result on finite jet determination and the first result of this kind in the infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p. This result is optimal since the automorphisms of the unit sphere are not determined by their 1-jets at a point of the sphere. The finite type condition is necessary since otherwise the needed jet order can be arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example, that determination by 2-jets fails for finite type hypersurfaces already in $\mathbb{C)$3.We also give an application to the dynamics of germs of local biholomorphisms of $\mathbb{C)$2.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/7014df777476744b91fcbe7c42e833f8af90548f

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In Algebra and Number Theory

1. B. Lamel
2. P. Ebenfelt
3. D. Zaitsev

In Applied Mathematics

1. D. Zaitsev
2. B. Lamel
3. P. Ebenfelt
4. Nordine Mir
5. Xiaojun Huang
6. M. S. Baouendi
7. W. Yin

Citations in the corpus

• Degenerate real hypersurfaces in $\mathbb {C}^2$ with few automorphisms (2006, methodology): “Theorem 1 can be viewed as a refinement of the results in [5] taking into account the finer invariants introduced in this paper.”
• Parametrization of local CR automorphisms by finite jets and applications (2006, background): “In dimension N = 2, the stability group of such real hypersurfaces has recently been studied by Ebenfelt, Zaitsev and the first author in [22], where it was shown that their local CR automorphisms are analytically parametrized by their 2-jets.”