• 제목/요약/키워드: Levi form of the boundary

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GEOMETRIC CHARACTERIZATION OF q-PSEUDOCONVEX DOMAINS IN ℂn

  • Khedhiri, Hedi
    • 대한수학회보
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    • 제54권2호
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    • pp.543-557
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    • 2017
  • In this paper, we investigate the notion of q-pseudoconvexity to discuss and describe some geometric characterizations of q-pseudoconvex domains ${\Omega}{\subset}{\mathbb{C}}^n$. In particular, we establish that ${\Omega}$ is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any (n-q+1)-dimensional subspace $E{\subset}{\mathbb{C}}^n$. Furthermore, we prove that the Kiselman's minimum principal holds true for all q-pseudoconvex domains in ${\mathbb{C}}^p{\times}{\mathbb{C}}^n$ such that each slice is a convex tube in ${\mathbb{C}}^n$.

A NOTE ON THE BOUNDARY BEHAVIOUR OF THE SQUEEZING FUNCTION AND FRIDMAN INVARIANT

  • Kim, Hyeseon;Mai, Anh Duc;Nguyen, Thi Lan Huong;Ninh, Van Thu
    • 대한수학회보
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    • 제57권5호
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    • pp.1241-1249
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    • 2020
  • Let Ω be a domain in ℂn. Suppose that ∂Ω is smooth pseudoconvex of D'Angelo finite type near a boundary point ξ0 ∈ ∂Ω and the Levi form has corank at most 1 at ξ0. Our goal is to show that if the squeezing function s(𝜂j) tends to 1 or the Fridman invariant h(𝜂j) tends to 0 for some sequence {𝜂j} ⊂ Ω converging to ξ0, then this point must be strongly pseudoconvex.

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

  • Cho, Sanghyun
    • 대한수학회지
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    • 제50권3호
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    • pp.479-491
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    • 2013
  • Let M be a smooth real hypersurface in complex space of dimension $n$, $n{\geq}3$, and assume that the Levi-form at $z_0$ on M has at least $(q+1)$-positive eigenvalues, $1{\leq}q{\leq}n-2$. We estimate solutions of the local $\bar{\partial}$-closed extension problem near $z_0$ for $(p,q)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

EXPLICIT SOBOLEV ESTIMATES FOR THE CAUCHY-RIEMANN EQUATION ON PARAMETERS

  • Cho, Sang-Hyun;Choi, Jae-Seo
    • 대한수학회보
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    • 제45권2호
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    • pp.321-338
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    • 2008
  • Let $\bar{M}$ be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures $\{L^{\tau}\}_{{\tau}{\in}I}$, $0{\in}I$, which extend smoothly up to bM, the boundary of M, and assume that there is ${\lambda}{\in}C^{\infty}$(bM) which is strictly subharmonic with respect to the structure $L^0|_{bM}$ in any direction where the Levi-form vanishes on bM. We obtain explicit estimates for the $\bar{\partial}$-Neumann problem in Sobolev spaces both in space and parameter variables. Also we get a similar result when $\bar{M}$ is strongly pseudoconvex.