Let Ω ⊂ C n × R be a bounded domain with smooth boundary such that ∂Ω has only nondegenerate elliptic CR singularities, and let f : ∂Ω → C be a smooth function that is CR at CR points of ∂Ω (when n = 1 we require separate holomorphic extensions for each real parameter). Then f extends to a smooth CR function on Ω, that is, an analogue of Hartogs-Bochner holds. In addition, if f and ∂Ω are real-analytic, then f is the restriction of a function that is holomorphic on a neighborhood of Ω in C n+1 . An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of ∂Ω as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.
Abstract. Let M ⊂ C n+1 , n ≥ 2, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every realanalytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of M . Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let Ω ⊂ C n × R, n ≥ 2, be a bounded domain with a connected real-analytic boundary such that ∂Ω has only nondegenerate CR singularities. We prove that if f : ∂Ω → C is a real-analytic function that is CR at CR points of ∂Ω, then f extends to a holomorphic function on a neighborhood of Ω in C n × C.
Let D be a domain in C n with smooth (that is, C ∞ ) boundary. If 0 ≤ α ≤ ∞ we denote by A α (D) the space of functions holomorphic on D and of class C α on D. We write A(D) for A 0 (D) and A ω (D) for the space of functions holomorphic on a neighborhood of D. We say that a point p ∈ ∂D is a peak point relative to D for A α (D) if there exists a function f ∈ A α (D) so that f (p) = 1 and |f | < 1 on D \ {p}. We call f a peak function. This condition is clearly equivalent to the existence of a strong support function, that is, a function g ∈ A α (D) so that g(p) = 0 and Re g > 0 on D \ {p}. We say that p ∈ ∂D is a local peak point forWe want to determine whether boundary points are peak points. Finding a peak function can be thought of as giving a quantitative converse to the maximum modulus principle. Now observe that if f is a peak function at p relative to D then 1/(1 − f ) is a holomorphic function on D with no holomorphic extension past p. Thus if every boundary point of D is a peak point then D is a domain of holomorphy. In light of this fact, if we want to determine whether boundary points are peak points it makes sense to restrict our attention to domains of holomorphy. By the solution of the Levi problem these are the (Levi) pseudoconvex domains. Here is another observation: If there is a complex disc in the boundary of D then (by the maximum modulus principle) no point in the relative interior of that disc can be a peak point for A(D). A natural condition in this context is that D be of finite type at the point p in question, in the sense of D'Angelo: There is a finite upper bound on the order of contact of nontrivial complex analytic varieties with the boundary at p. The infimum of all such upper bounds is called the type at p. We formulate the major open question in terms of this notion.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.