Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Lebl, Jiri; Noell, Alan; Ravisankar, Sivaguru

doi:10.1007/s12220-017-9767-6

Cited by 4 publications

(9 citation statements)

References 23 publications

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“…where f k is a real-homogeneous polynomial of degree k and f is O(k + 1). The rest of the proof is essentially identical to the proof of Proposition 5.1 from [23]. Let us go through it quickly.…”

Section: Formal Extension At a Cr Singularitymentioning

confidence: 87%

“…Let M c ′ ⊂ C × C be the manifold defined by the pullback The lemma implies the existence of a formal power series for an extension. The following proposition and its proof is essentially the same as Proposition 5.1 from [23] with the necessary modifications made for smooth functions rather than real-analytic functions. Proposition 6.2.…”

Section: Formal Extension At a Cr Singularitymentioning

confidence: 97%

“…In this paper we address what happens in the nondegenerate smooth case. Via the results of [23], a formal extension always exists, but a smooth extension does not exist in all cases.…”

Section: Introductionmentioning

confidence: 99%

“…Additionally we also use natural complex manifolds with boundary attached to M to show existence of the extension for s > 0. In §6 we use the results of [23] to show that a formal extension exists at CR singularities. We prove that the extension is regular up to the boundary in §7.…”

Section: Introductionmentioning

confidence: 99%

“…The authors considered the extension of CR functions for hypersurfaces of C n × R in the elliptic case [22], and in the general nondegenerate real-analytic case [23]. In this paper we address what happens in the nondegenerate smooth case.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Formal Extension At a Cr Singularitymentioning

confidence: 87%

Section: Formal Extension At a Cr Singularitymentioning

confidence: 97%

“…In this paper we address what happens in the nondegenerate smooth case. Via the results of [23], a formal extension always exists, but a smooth extension does not exist in all cases.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

Lebl

Noell

Ravisankar

2018

Trans. Amer. Math. Soc.

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A CR singular analogue of Severi’s theorem

2021

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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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Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Cited by 4 publications

References 23 publications

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ

A CR singular analogue of Severi’s theorem

On the Levi-flat Plateau problem

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