On partial analyticity of CR mappings

We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions. Received November 26, 1998; in final form March 23, 1999 / Published online October 11, 2000     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

Determination of formal CR mappings by a finite jet

We prove that if M is a connected real-analytic holomorphically nondegenerate hypersurface in Cn+1, then for any point p∈M there exists an integer k such that any two germs at p of local biholomorphic mappings that send M into itself and whose k-jets agree at p are identical.The above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose Jacobian determinant does not vanish identically.     

Regularity of CR mappings

Sloan Fellow, partially supported by NSF grant DMS 8619858     

Some regularity theorems for CR mappings

The purpose of this article is to study Lipschitz CR mappings from an h-extendible (or semi-regular) hypersurface in ${\mathbb{C}^n}$ . Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.     

Functor of weakly additive τ-smooth functionals and mappings

We present a criterion for the τ-smoothness of weakly additive order-preserving functionals and establish that the functor of τ-smooth weakly additive functionals preserves the perfectness of mappings. We prove that if a continuous mapping between Tikhonov spaces is such that the mapping between the corresponding spaces of weakly additive functionals generated by it is open, then the indicated mapping is also open. It is shown that the converse statement is true under certain additional conditions.     

Conformal mappings and CR mappings on the Engel group

We show that conformal mappings between the Engel groups are CR or anti-CR mappings. This reduces the determination of conformal mappings to a problem in the theory of several complex analysis. The result about the group of CR automorphisms is used to determine the identity component of the group of conformal mappings on the Engel group.     

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M⊂CN, N?2, that is essentially finite and of finite type at each of its points, for every point p∈M there exists an integer ?p, depending upper-semicontinuously on p, such that for every smooth generic submanifold M^′⊂CN of the same dimension as M, if are two germs of smooth finite CR mappings with the same ?p jet at p, then necessarily for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω,Ω^′⊂CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p∈∂Ω and agreeing up to order k at p, then necessarily H₁=H₂.     

Number-theoretic properties of certain CR mappings

We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings lead to a collection of polynomials ƒ_p,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some surprising number-theoretic results about the ƒ_p,q, in particular, ƒ_p,q is congruent to x^P + y^P modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3 and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for certain representations of dihedral groups to illustrate the non-Abelian case.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

Regularity of CR mappings between algebraic hypersurfaces

Oblatum 8-VI-1995 & 18-X-1995     

Slow mappings of finite distortion

We examine mappings of finite distortion from Euclidean spaces into Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.     

On the unique continuation problem for CR mappings into nonminimal hypersurfaces

We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ? ⁿ . If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp ₀ εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′₀ =H(p ₀ (i.e.,M′ is nonminimal atp′ ₀), andH(M) ? Σ′ (forcingM to be nonminimal atp ₀), then the transversal component ofH is not flat atp ₀. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ? ?², real analytic and of infinite type atp ₀ andp′₀ respectively (without being Levi flat), such thatH(M) ? Σ′ but the transversal component ofH is flat atp ₀ (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ? Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.     

Analytic sets and the boundary regularity of CR mappings

It is shown that if a continuous CR mapping between smooth real analytic hypersurfaces of finite type in extends as an analytic set, then it extends as a holomorphic mapping.