Analytic regularity of CR maps into spheres

Abstract

Abstract. Let M ⊂ C N be a connected real-analytic hypersurface and S 2N −1 ⊂ C N the unit real sphere, N > N ≥ 2. Assume that M does not contain any complex-analytic hypersurface of C N and that there exists at least one strongly pseudoconvex point on M . We show that any CR map f : M → S 2N −1 of class C N −N +1 extends holomorphically to a neighborhood of M in C N .

Unique jet determination and extension of germs of CR maps into spheres

“…It follows from Theorem 1.4 that f extends holomorphically along any path in M starting from p 0 . Hence, since M is connected and simply-connected, we can extend, by analytic continuation, the local map f holomorphically to a neighborhood of M in C N .The second part of Corollary 1.5 follows from the first part of it together with the regularity results in[25,41].…”

“…Let Ω ⊂ C N be a bounded domain with smooth real-analytic boundary and B N ′ ⊂ C N ′ be the unit ball. Then there exists an integer ℓ, depending only on ∂Ω, such that if F ,G : Ω → B N ′ are two proper holomorphic mappings extending smoothly up to the boundary near some point p ∈ ∂Ω with j ℓ p F = j ℓ p G, it follows that F = G. We will establish Theorem 1.1 (as well as Corollaries 1.2 and 1.3) for local holomorphisms, since all C ∞ -smooth CR maps under consideration automatically extend holomorphically to a neighborhood of p in C N according to [40,41].…”

Holomorphic extension of meromorphic mappings along real analytic hypersurfaces

“…When the real hypersurfaces lie in the complex spaces of same dimension, CR mappings of given smoothness must be real-analytic, (see for example [1]). In the case of real hypersurfaces of different dimensions, analyticity of CR mappings with given smoothness on the boundary was shown provided that the target is a real sphere (see for example [2,7] ). In the proof, they first show that the CR mappings extend meromorphically.…”

On the analyticity of CR-diffeomorphisms

“…The reflection principle for merely continuous CR-maps between real-analytic hypersurfaces which are of D'Angelo finite type (meaning they do not contain any complex varieties) in C N , N ≥ 3, is contained in the work of Diederich and Pinchuk [17]. For notable results on the reflection principle for CR-mappings between CR-submanifolds of different dimension see Coupet, Pinchuk and Sukhov [15], Meylan, Mir and Zaitsev [39] and Mir [40]. However, these positive results do not apply to more degenerate situations, and also do not help to shed light on the different roles of minimality and nondegeneracy.…”