Local complex foliation of real submanifolds

Freeman, Michael

doi:10.1007/bf01432883

Cited by 54 publications

(25 citation statements)

References 3 publications

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“…Moreover, under this constant rank assumption on N 1,0 M there is a unique smooth foliation of M ∩U by complex manifolds such that the complex tangent space to the leaf passing through z∈M is N 1,0 M (see [7]). Now we return to the main problem.…”

Section: Lemma 32mentioning

confidence: 98%

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Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Sargsyan

2012

Ark. Mat.

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Section: Lemma 32mentioning

confidence: 98%

“…, there exists a subsequence, which we again denote by f k , which converges weakly to some f in L 2 n,q (X α , N, 0) such that f k →f strongly on a common exceptional set K 2 of the basic estimate (7). As…”

Section: Lemma 32mentioning

confidence: 98%

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Sargsyan

2012

Ark. Mat.

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“…As in the convex case (see Theorem 6.21) we can find a complex foliation in some parts of the Shilov boundary for q-plurisubharmonic functions on smoothly bounded domains. For further results on complex foliations of real submanifolds we refer to [Fre74]. of the Shilov boundary for PSH q (D) in bD locally admits a foliation by complex qdimensional submanifolds, provided it is not empty.…”

Section: Lemma 84mentioning

confidence: 99%

Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Pawlaschyk

2016

Ann. Polon. Math.

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We introduce a notion of the Shilov boundary for some subclasses of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subclasses with simple structure one can show the existence and uniqueness of the Shilov boundary. Then we provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of Bychkov's theorem which gives a geometric characterization of the Shilov boundary for q-plurisubharmonic functions on convex bounded domains. In the case of bounded pseudoconvex domains with smooth boundary we also show that some parts of the Shilov boundary for q-plurisubharmonic functions are foliated by q-dimensional complex submanifolds.

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“…Assume that D is of finite type (as defined in Section 1), or more generally that ∂D contains no nontrivial complex manifold. It turns out that the set of boundary points at which D is strictly pseudoconvex is dense in the boundary of D. (If not, by Freeman's work in [28] there would be a local foliation of the boundary by complex submanifolds, a contradiction. For a precise statement in the case of finite type see the Main Theorem of Catlin's paper [9].)…”

Section: Theoremmentioning

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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Local complex foliation of real submanifolds

Cited by 54 publications

References 3 publications

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Peak Points for Pseudoconvex Domains: A Survey

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