The Levi form and local complex foliations

Freeman, Michael

doi:10.1090/s0002-9939-1976-0409899-2

Cited by 14 publications

(10 citation statements)

References 4 publications

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“…We are interested in knowing whether S might inherit any complex structure from C", that is we are interested in knowing whether there is a foliation 9 of S by manifolds that are complex submanifolds of C". A well known approach to this problem is to show that the Levi distribution LS is integrable, see Sommer [ll] and Freeman [ 5 ] . In a certain sense the Levi foliation is the smallest possible foliation one can look for.…”

Section: The Levi Foliationmentioning

confidence: 99%

Foliations and complex monge‐ampère equations

Bedford

Kalka

1977

Comm Pure Appl Math

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Here we shall study some nonlinear partial differential equations involving the complex hessian cfic) = (d2f/dz, a&) which are related to the theory of functions of several complex variables. Since 8 and a are defined independently of the choice of local coordinates, the forms in the equations are invariant under holomorphic mappings. Certain properties of the forms on the left-hand side of (l.l), (1.2), and (1.3) were discussed by Chern, Levine, and Nirenberg [4] for the case where f is a bounded, real, plurisubharmonic function. The approach here is to drop the assumption that f is real and plurisubharmonic and require instead that f is Y 3 . In C", the (1, 1)-form %f may be identified with the complex hessian (hi) so that (l.l), (1.2), and (1.3) are easily seen to be equivalent (respectively) to the following three conditions (1.1y rank c f i~) 5 p ,

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Section: The Levi Foliationmentioning

confidence: 99%

Foliations and complex monge‐ampère equations

Bedford

Kalka

1977

Comm Pure Appl Math

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“…This completes the proof of Lemma 3. We remark that a similar use of the Jacobi identity occurs in [15].…”

Section: Suppose That We Can Show That [J (A) A] Is a Linear Combinamentioning

confidence: 72%

“…Thus the commutator we are interested in is X 1 , X 1 = 2i[B, J (B)]. If X 1 were a Levi null field in a full neighborhood of P , then [X 1 , X 1 ] would likewise be ( [15]). However, we only know that L(X 1 , X 1 ) = 0 on M. What we can assert is that…”

Section: Suppose That We Can Show That [J (A) A] Is a Linear Combinamentioning

confidence: 99%

“…Define m to be the maximum rank of the Levi form on V and let P ∈ V be a point where the Levi form has rank m (such a point exists, since the rank assumes only finitely many values on V ). Then near P , the rank is at least m, hence equal to m. Therefore, bΩ is foliated, near P , by complex manifolds of dimension n − 1 − m (see for example [15]). Theorem 1 (for k = n − 1 − m) implies that the ∂-Neumann operator on Ω is not compact.…”

Section: Lemmamentioning

confidence: 99%

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Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

Şahutoğlu

Straube

2006

Math. Ann.

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We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain in C n is an obstruction to compactness of the ∂-Neumann operator on , provided that at some point of M, the Levi form of b has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. (2000): 32W05 Mathematics Subject Classification

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“…Therefore, the dimension of the null space of the Levi form is constant at every y ∈ U x ∩bΩ and satisfies dim N y ≥ q. The foliation result of Freeman and Sommer ([19], [20], [9], [10]), which holds regardless of pseudoconvexity, implies the open set U x ∩ bΩ in the boundary of the domain is foliated by complex manifolds of dimension equal to this constant dimension of the null space of the Levi form, which is at least q. This violates finite D'Angelo q-type that is assumed to hold on all of Ũ ∩ bΩ.…”

Section: Remarksmentioning

confidence: 99%

Effective vanishing order of the Levi determinant

Nicoara

2011

Math. Ann.

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On a smooth domain in complex n space of finite D'Angelo q-type at a point, an effective upper bound for the vanishing order of the Levi determinant $\text{coeff}\{\partial r \wedge \dbar r \wedge (\partial \dbar r)^{n-q}\}$ at that point is given in terms of the D'Angelo q-type, the dimension of the space n, and q itself. The argument uses Catlin's notion of a boundary system as well as techniques pioneered by John D'Angelo.Comment: 22 pages; typos in example from p.20 fixed in the second versio

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The Levi form and local complex foliations

Abstract: Abstract. A short coordinate-free proof is given for some known results on the existence of local complex-analytic foliations of a real submanifold M of C". The proof uses an explicit formulation of the equivalence between two definitions of the E. E. Levi form of M.

Cited by 14 publications

References 4 publications

Foliations and complex monge‐ampère equations

Foliations and complex monge‐ampère equations

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

Effective vanishing order of the Levi determinant

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