Extension of CR functions from boundaries in Cn x R

Lebl, Jiri; Noell, Alan; Ravisankar, Sivaguru

doi:10.1512/iumj.2017.66.6067

Cited by 8 publications

(17 citation statements)

References 20 publications

(25 reference statements)

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“…The map F −1 is continuous on F (Ω), so F −1 | F (∂Ω) extends continuously along each leaf in F (Ω) to a holomorphic map. Because F (∂Ω) has only elliptic singularities, by the main result of [11], F −1 | F (∂Ω) extends to a realanalytic CR map on F (Ω). As in the proof of Lemma 6.5, it follows that the derivative of F is nonsingular at each CR singular point.…”

Section: Flat Case In Cr Dimensionmentioning

confidence: 97%

“…Proof. Note that the condition on f is precisely what is required in the main result of [11] to obtain an extension F : Ω → C 2 that is real-analytic and CR. By Lemma 6.1, F maps into C × R.…”

Section: Flat Case In Cr Dimensionmentioning

confidence: 99%

“…In this paper we prove the regularity of this solution for real-analytic M . For n = 1, [11] provides existence in the elliptic case under a hypothesis replacing the CR condition. We prove regularity in this case if the image is flat.…”

Section: Introductionmentioning

confidence: 99%

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On the Levi-flat Plateau problem

Lebl¹,

Noell²,

Ravisankar³

2020

Complex Anal Synerg

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We solve the Levi-flat Plateau problem in the following case. Let M ⊂ C n+1 , n ≥ 2, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose M is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in C n × R with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary M . We also study boundary regularity of CR automorphisms of domains in C n × R.

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Section: Flat Case In Cr Dimensionmentioning

confidence: 97%

Section: Flat Case In Cr Dimensionmentioning

confidence: 99%

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On the Levi-flat Plateau problem

Lebl¹,

Noell²,

Ravisankar³

2020

Complex Anal Synerg

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“…When n = 1, the CR condition is vacuous, and every smooth function is a CR function. An extra condition is necessary for extension, and under an extra condition the authors proved an extension in the elliptic case in [22].…”

Section: Introductionmentioning

confidence: 99%

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

Lebl

Noell

Ravisankar

2018

Trans. Amer. Math. Soc.

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We prove an analogue of the Lewy extension theorem for a real dimension 2n smooth submanifold M ⊂ C n × R, n ≥ 2. A theorem of Hill and Taiani implies that if M is CR and the Levi-form has a positive eigenvalue restricted to the leaves of C n × R, then every smooth CR function f extends smoothly as a CR function to one side of M . If the Levi-form has eigenvalues of both signs, then f extends to a neighborhood of M . Our main result concerns CR singular manifolds with a nondegenerate quadratic part Q. A smooth CR f extends to one side if the Hermitian part of Q has at least two positive eigenvalues, and f extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed. M = n−1, but dim T (0,1) p M = n is possible. If dim T (0,1) p M = n for all p, then M is a complex submanifold by the Newlander-Nirenberg theorem, and so it is locally equal to C n × {s 0 } for some s 0 , and f extends to both sides of M if and only if f is holomorphic on M. Thus, assume that at least somewhere dim T (0,1) pThe points where dim T (0,1) p M = n − 1 are called CR points of M, and the points where dim T (0,1) p M = n are the so-called CR singularities. Write M CR for the set of CR points of M. We need a definition of a CR function on a possibly CR singular submanifold, and we take the definition in the weakest possible sense: A function f : M → C is CR, if vf = 0 for all CR vector fields on M CR . Alternatively, we obtain the same definition if we allow v to be smooth vector fields on M such that v p ∈ T (0,1) p

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“…For either a degenerate CR singularity or a non-flat CR singularity counterexamples exist, see below. The authors have studied the smooth case of this problem in [20] for elliptic CR singularities.…”

Section: Introductionmentioning

confidence: 99%

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

2017

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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.

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Extension of CR functions from boundaries in Cn x R

Cited by 8 publications

References 20 publications

On the Levi-flat Plateau problem

On the Levi-flat Plateau problem

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

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