The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

Bharali, Gautam

doi:10.1007/s00209-011-0902-y

Cited by 3 publications

(6 citation statements)

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“…This type of CR singularity is called degenerate CR singularity. Before proceeding further with the discussion, we mention the following definition on nonparabolic CR singularity [6,13], which makes sense in case of degenerate CR singularity as well as nondegenerate.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The question arises: Is it possible to characterize the local polynomial convexity of a surface with nonparabolic CR singularity of oder k, k ≥ 3? Efforts [6,13] have been made to achieve a Bishop-type dichotomy for nonparabolic points in case of higher order CR singular points, but one of the assumptions of the results in this directions is, up to a biholomorphic change of variables, the lowest order homogeneous term is real valued. Maslov-type index (see Subsection 2.1 for definition) plays a crucial role in case of higher order degeneracy (see [6]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Efforts [6,13] have been made to achieve a Bishop-type dichotomy for nonparabolic points in case of higher order CR singular points, but one of the assumptions of the results in this directions is, up to a biholomorphic change of variables, the lowest order homogeneous term is real valued. Maslov-type index (see Subsection 2.1 for definition) plays a crucial role in case of higher order degeneracy (see [6]). The papers [4,5] describe conditions, in terms of the coefficients bound of the lowest order homogeneous term (not necessarily real valued), under which M is locally polynomially convex at the origin.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Forstnerič-Stout [9] used this to show local polynomial convexity at hyperbolic CR singularity. Approaching to local polynomial convexity at CR singularity of higher order, in general, is difficult; this is the main reason of assuming 'thin' or 'flat' surfaces, i.e., the lowest order homogeneous terms in the graphing function to be real valued in [13,6]. In [4,5], pulling back by proper holomorphic mapping on C 2 to union of certain totally-real surfaces is used crucially.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…such that ∂f ∂z Next, we mention a lemma due to Bharali [6], which gives an easy way to compute the Maslov-type index of a graph of homogeneous polynomial in z and z of degree k around an isolated CR singularity at the origin.…”

Section: Preliminariesmentioning

confidence: 99%

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Certain smooth real surfaces in $\mathbb{C}^2$ with singularity

Gorai

2019

Preprint

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Under certain geometric condition, the surfaces in C 2 with isolated CR singularity at the origin and with cubic lowest degree homogeneous term in its graph near the origin, can be reduced, up to biholomorphism of C 2 , to a one parameter family of the formnear the origin. We prove that Mt is not locally polynomially convex if t < 1. The local hull contains a ball centred at the origin if t < √ 3/2. We also prove that Mt is locally polynomially convex for t ≥ 15 − √ 33 2 √ 2 = 1.076.... We show that, for √ 3/2 ≤ t < 1, the local hull of Mt contains a one parameter family of analytic discs passing through the origin. We also show that local polynomial convexity of the union of finitely many pairwise transverse totally-real submanifolds of C n at the origin (their intersection) implies local polynomial convexity of the union of their sufficiently small C 1 -perturbation at their intersection, the origin. Some new results about the local polynomial convexity of the union of three totally-real planes are also reported.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%