Nondefective Stationary Discs and 2-Jet Determination in Higher Codimension
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Nondefective Stationary Discs and 2-Jet Determination in Higher Codimension Florian Bertrand1 · Francine Meylan2 Received: 8 January 2020 / Accepted: 5 September 2020 © Mathematica Josephina, Inc. 2020
Abstract We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of C N of class C4 . Keywords Stationary discs · Defect of analytic discs · Jet determination of CR automorphisms Mathematics Subject Classification 32V40 · 32H02
1 Introduction In his important paper [30,31], Tumanov introduced the notion of defect of an analytic disc f attached to a generic submanifold M ⊂ C N , which was defined equivalently by Baouendi et al. [1] as the dimension of the real vector space of all holomorphic lifts f in T ∗ (C N ) of f attached to the conormal bundle N ∗ (M). In particular, Tumanov proved that the existence of nondefective analytic discs, that is of defect 0, attached to M implies the wedge extendability of CR functions. In the present paper, we introduce a new notion of nondegeneracy of the Levi map which expresses the existence of a nondefective stationary disc attached to the quadric model of M. Using a deformation argument developed by Forstneriˇc [16] and Globevnik [17], we produce a family of stationary discs near that nondefective disc, which are uniquely determined by their 1-jet and which cover an open set of M. As an application of this theory, we deduce a 2-jet determination for CR automorphisms of our generic submanifold M.
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Florian Bertrand [email protected] Francine Meylan [email protected]
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Department of Mathematics, American University of Beirut, Beirut, Lebanon
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Department of Mathematics, University of Fribourg, Perolles, 1700 Fribourg, Switzerland
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F. Bertrand, F. Meylan
Theorem 1.1 Let M ⊂ C N be a C4 generic real submanifold. We assume that M is D-nondegenerate at p ∈ M. Then any germ at p of CR automorphism of M of class C3 is uniquely determined by its 2-jet at p. We refer to Definition 2.3 for the notion of D-nondegenerate submanifold; this notion is closely related to the existence of a nondefective stationary disc (see Lemma 3.3). In the previous paper by the authors and Blanc–Centi [6], 2-jet determination is obtained under the more restrictive assumption that M is fully nondegenerate (see Definition 1.2 in [6]). Indeed, while fully nondegeneracy imposes the codimension restriction d ≤ n, D-nondegeneracy requires d ≤ 2n. For instance, the quadric ⎧ 2 ⎪ ⎨ew1 = |z 1 | ew2 = |z 2 |2 ⎪ ⎩ ew3 = z 1 z 2 + z 1 z 2 and its perturbations are D-nondegenerate but not fully nondegenerate. Stationary discs were introduced by Lempert [24] in his work on the Kobayashi metric of strongly convex domains and studied further in [18,27,28,32]. Their use in the finite jet determination problem in the framework of finitely smooth submanifolds has been recently developed by several authors [5–7,33]. Finite jet determination problems for real analytic or C∞ real submanifolds has attrac
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