Convergent normal form and canonical connection for hypersurfaces of finite type in C2

Abstract

We study the holomorphic equivalence problem for finite type hypersurfaces in C 2 . We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in C 2 .

Normal forms in Cauchy-Riemann geometry

“…In addition, a convergent normal form is important for understanding the geometry of weakly pseudoconvex domains, which is still far from being understood completely in Several Complex Variables. The convergence problem for the formal normal form in [Kol05] was addressed in the work [KZ14], [KZ15] of the second and the third author. The difficulties in studying the convergence can be roughly described as follows.…”

“…In the second case, when there exists a two-dimensional surface consisting of points of constant type, the strategy of the convergence proof is very different. In this case, a formal normal form (which is slightly different in [KZ15] from the one in [Kol05]) provides one with a distinguished direction field in the Levi-degeneracy set Σ. Namely, at each point q ∈ Σ one takes the direction of the vector which is transformed into the distinguished direction (2.13) in some (and hence any) normal form at q. This gives one a distinguished foliation in Σ, and each of the leaves is called now a degenerate chain.…”

“…In [KZ14], the authors use the strategy of Chern-Moser operators to obtain a formal normal form with the convergence property. The proof of the convergence there is somehow similar to that for Theorem 6, that is why we omit discussing the formulation and the proof here (in fact, the paper [KZ14] preceded the paper [KZ15] where Theorem 6 was proved, however, we chose Theorem 6 for the detailed description as it is technically simpler).…”

“…The main difficulty here is that the known convergent constructions [KZ14], [KZ15] are based on the presence of an entire curve passing through the reference finite type point and having certain geometric data constant along it. No such curve exists at an isolated weakly pseudoconvex point.…”

The equivalence theory for infinite type hypersurfaces in $\mathbb C^2$

“…It also provides a solution to the CR equivalence problem, because every CR diffeomorphism of two real-analytic hypersurfaces of finite type in C 2 extends to a biholomorphic map by a result of Baouendi, Jacobowitz, and Treves [2]. Kolář's normal form has been shown to be convergent under certain geometric conditions, see the work [33] of Kossovskiy-Zaitsev, but is divergent in general [26]. For Levi-degenerate hypersurfaces in C N , N ≥ 3 satisfying certain special conditions (in addition to the Hormander-Kohn bracket-generating condition) normal form constructions were carried out by Ebenfelt [16,15] and by Kossovskiy-Zaitsev [33].…”