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 $\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.