Pseudoconvex Domains with Real-Analytic Boundary

Diederich, Klas; Fornæss, John Erik

doi:10.2307/1971120

Cited by 211 publications

(117 citation statements)

References 8 publications

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“…This, however, implies that the compact, real-analytic variety X contains the germ of an analytic variety, in contradiction to a result of Diederich and Fornaess [6]. The proof of Theorem 5 is complete.…”

Section: For Each (T ζ) ∈ [−τ O τ O ] × T the Point ϕ(T ζ) Lies mentioning

confidence: 73%

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Stout

2006

Proc. Amer. Math. Soc.

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Section: For Each (T ζ) ∈ [−τ O τ O ] × T the Point ϕ(T ζ) Lies mentioning

confidence: 73%

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Stout

2006

Proc. Amer. Math. Soc.

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“…Through each point of M CR there exists a germ of a complex variety of complex dimension n − 1 contained in M. The set of CR points is dense in M. Take a sequence p k of CR points converging to the origin and take complex varieties of dimension n − 1, W k ⊂ M with p k ∈ W k . A theorem of Fornaess (see Theorem 6.23 in [27] for a proof using the methods of Diederich and Fornaess [11]) implies that there exists a variety through W ⊂ M with 0 ∈ W and of complex dimension at least n − 1.…”

Section: Levi-flat Quadricsmentioning

confidence: 99%

Normal forms for CR singular codimension-two Levi-flat submanifolds

Gong

Lebl

2015

Pacific J. Math.

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Abstract. Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to C m , m > 2, of Bishop surfaces in C 2 . Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of z m =z 1 z 2 +z 2 1 , have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in C 3 in the nondegenerate case that shows infinitely many formal invariants.

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“…These neighborhoods are constructed using a modification of the technique introduced by Diederich and Fornaess in [16]. Their technique requires the existence of a certain stratification of the boundary, and in the case of D such a stratification exists by the work of Diederich and Fornaess in [17].…”

Section: Sketch Of Proofmentioning

confidence: 99%

Peak Points for Pseudoconvex Domains: A Survey

Noell

2008

J Geom Anal

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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.

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Pseudoconvex Domains with Real-Analytic Boundary

Cited by 211 publications

References 8 publications

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Holomorphic approximation on compact, holomorphically convex, real-analytic varieties

Normal forms for CR singular codimension-two Levi-flat submanifolds

Peak Points for Pseudoconvex Domains: A Survey

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