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

Abstract

We show that germs of local real-analytic CR automorphisms of a real-analytic hypersurface M in C 2 at a point p ∈ M are uniquely determined 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 C 3 .We also give an application to the dynamics of germs of local biholomorphisms of C 2 .

Mappings between real submanifolds in complex space

“…For a hypersurface in C 2 Ebenfelt, Lamel & Zaitsev [ELZ01] have recently found optimal results for finite determination of biholomorphisms. …”

A hypersurface in $\mathbb {C}^2$ whose stability group is not determined by $2$-jets

“…(For more information, see the survey articles [BER00] and [Vit90].) More recently, Ebenfelt, Lamel, and Zaitsev [ELZ00] have shown that the stability group of any hypersurface of finite type in C 2 is determined by 2-jets.…”

“…Hence, the stability groups of hypersurfaces of infinite type need not be determined by 2-jets. However, in [ELZ00] it is shown that the stability group of any nonflat realanalytic hypersurface is determined by jets of some predetermined finite order. For the special case of so-called 1-infinite type hypersurfaces, of which M is an example, the author has shown [Kow01] that the stability group is in fact formally parametrized by such a finite jet.…”

On a Family of Hypersurfaces in C2 with Stability Groups Determined by High Jet-Order

“…These results have been extended more recently to ever more general hypersurfaces. For example, Ebenfelt, Lamel, and Zaitsev [5] proved the stability group of any Levi nonflat hypersurface (M, p) in C 2 is determined by k-jets for some finite number k; moreover, if M is of finite type at p (i.e. contains no complex hypersurface passing through p), then 2-jets will always suffice.…”