Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

Abstract

The classical Schwarz reflection principle states that a continuous map f between real-analytic curves M and M ′ in C that locally extends holomorphically to one side of M , extends also holomorphically to a neighborhood of M in C. It is well-known that the higher-dimensional analog of this statement for maps f : M → M ′ between real-analytic CR-submanifolds M ⊂ C andM ′ ⊂ C ′ does not hold without additional assumptions (unlessM andM ′ are totally real). In this paper, we assume that f is C-smooth and that the target M ′ is real-algebraic, i.e. contained in a real-algebraic subset of the same dimension. If f is known to be locally holomorphically extendible to one side of M (when M is a hypersurface) or to a wedge with edge M (when M is a generic submanifold of higher codimension), then f automatically satisfies the tangential CauchyRiemann equations, i.e. it is CR. On the other hand, if M is minimal, any CR-map f : M → M ′ locally extends holomorphically to a wedge with edge M by Tumanov’s theorem [Tu88] and hence, in that case, the extension assumption can be replaced by assuming f to be CR. Local holomorphic extension of a CR-map f : M → M ′ may clearly fail when M ′ contains an (irreducible) complex-analytic subvariety E ′ of positive dimension and f(M) ⊂ E . Indeed, any nonextendible CR-function on M composed with a nontrivial holomorphic map from a disc in C into E ′ yields a counterexample. Our first result shows that this is essentially the only exception. Denote by E ′ the set of all points p ∈ M ′ through which there exist irreducible complex-analytic subvarieties of M ′ of positive dimension. We prove: Theorem 1.1. Let M ⊂ C and M ′ ⊂ C ′ be connected real-analytic and real-algebraic CRsubmanifolds respectively. Assume that M is minimal at a point p ∈ M . Then for any C-smooth CR-map f : M → M , at least one of the following conditions holds: (i) f extends holomorphically to a neighborhood of p in C ; (ii) f sends a neighborhood of p in M into E . If M ′ is a real-analytic hypersurface, the set E ′ consists exactly of those points that are not of finite type in the sense of D’Angelo [D82] (see Lempert [L86] for the proof) and, in particular,