On the Polynomial Convexity of the Union of Three Totally Real Planes in ℂ2

Gorai, Sushil

doi:10.1093/imrn/rns200

Cited by 2 publications

(4 citation statements)

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“…Union of three totally-real planes in C 2 was first consider by Thomas [26,27] showing that there exist triples of totally-real planes such that the local hull of the union contains a ball centred at the origin and, also, by demonstrating examples where there are no nontrivial hull. [11] (see Result 2.7) deals extensively with the unions of three totally-real planes in C 2 . In this paper, we provide some new results in this setting (see Theorem 3.1, Theorem 3.2, Theorem 3.3) that will also be used in the proof of Theorem 1.8.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The next couple of lemmas are of linear algebraic flavour. The first one from [11] gives certain normal form for pair of matrices under similarity of matrices with real entries.…”

Section: Preliminariesmentioning

confidence: 99%

“…Next, we mention a general result from [11] about local polynomial convexity of the union of three totally-real planes in C 2 . It gives a sufficient condition, in terms of the matrices involved in the Weinstock's normal form, for local polynomial convexity of union of three totally-real planes at 0 ∈ C 2 .…”

Section: Preliminariesmentioning

confidence: 99%

“…It gives a sufficient condition, in terms of the matrices involved in the Weinstock's normal form, for local polynomial convexity of union of three totally-real planes at 0 ∈ C 2 . We need few notations from [11]:…”

Section: Preliminariesmentioning

confidence: 99%

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

Gorai

2019

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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%

“…The next couple of lemmas are of linear algebraic flavour. The first one from [11] gives certain normal form for pair of matrices under similarity of matrices with real entries.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations