Pseudoconvex Domains Research Articles

Pinchuk scaling method on domains with non-compact automorphism groups

In this paper, we characterize weakly pseudoconvex domains of finite type in [Formula: see text] in terms of the boundary behavior of automorphism orbits by using the scaling method.

The Gehring–Hayman type theorem on pseudoconvex domains of finite type in $$\mathbb {C}^2$$

The Gehring–Hayman type theorem on pseudoconvex domains of finite type in $$\mathbb {C}^2$$

Geometry of analytic continuation on complex manifolds – history, survey, and report

Abstract Beginning with the state of art around 1953, solutions of the Levi problem on complex manifolds will be recalled at first up to Takayama’s result in 1998. Then, the activity of extending the results by the L 2 {L}^{2} method in these decades will be reported. The method is by exploiting the finite dimensionality of certain L 2 {L}^{2} ∂ ¯ \bar{\partial } -cohomology groups to prove that a Hermitian holomorphic line bundle L L over a complex manifold M M is bimeromorphically equivalent to an ample bundle when it is restricted to a bounded locally pseudoconvex domain Ω ⋐ M \Omega \hspace{0.15em}\Subset \hspace{0.15em}M under the positivity of L ∣ ∂ Ω {L| }_{\partial \Omega } and the regularity of ∂ Ω \partial \Omega .

Global Newlander–Nirenberg Theorem for Domains with C2 Boundary

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander–Nirenberg theorem under the presence of a C2 strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with C2 boundary D⊂Cn, we show the existence of global holomorphic coordinate systems defined on D‾ that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small C2 perturbation of ∂D. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a C2 real hypersurface M⊂Cn, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.

Stability and localization of the Lp Bergman kernel

The aims of this paper are twofold. First, we generalize the classical Ramadanov theorem and Skwarczyński theorem for the [Formula: see text] Bergman kernels to the [Formula: see text] case, which are concerned with the compact convergence of [Formula: see text] Bergman kernels on an increasing or decreasing sequence of domains in [Formula: see text]. Second, we prove a localization principle for the [Formula: see text] Bergman kernel on bounded strongly pseudoconvex domains with smooth boundary. Utilize the localization, we study the asymptotic behavior of the [Formula: see text] Bergman kernel near strongly pseudoconvex boundary points and show that the [Formula: see text]-Schwarz content is identical to [Formula: see text] on a neighborhood of a strongly pseudoconvex boundary point.

A Gehring–Hayman Inequality for Strongly Pseudoconvex Domains

Abstract We prove that if $D$ is a strongly pseudoconvex domain with $\mathcal C^{2, \alpha }$-smooth boundary, then the length of a geodesic for the Kobayashi–Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the points.

The H2-Corona problem on delta-regular domains

We prove an H2-Corona theorem with estimate C(δ)=Cδ−1−q|log⁡δ| for δ≪1 on delta-regular domains, where q=min⁡{n,m−1} and m is the number of generators. This class of domains includes smooth bounded domains with defining functions that are plurisubharmonic on boundaries and pseudoconvex domains of D'Angelo finite type.

L2 ∂¯-cohomology with weights and bundle convexity of certain locally pseudoconvex domains

L2 ∂¯-cohomology with weights and bundle convexity of certain locally pseudoconvex domains

Splitting type results for pseudoconvex domains and remarks on their Nebenhülle

We give a very general splitting type theorem for biholomorphic maps close to identity in the context of smoothly bounded pseudoconvex domains (Theorem 1.4). As a particular case, in the context of worm domains, we essentially reprove the splitting type result (Theorem 1.3) from Bracci et al. (Math Z 292:879–893, 2019) (by a different method). We also discuss some properties of the Nebenhülle of worm domains.

Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reductions Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document} for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} labeling a unitary irreducible representation. There is a natural residual circle action on Xνred\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^{\ extrm{red}}_{\ u }$$\\end{document}. We have two natural decompositions of the corresponding Hardy spaces H(X) and H(Xνred)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })$$\\end{document}. The first one is given by the ladder of isotypes H(X)kν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X)_{k\ u }$$\\end{document}, k∈Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}$$\\end{document}; the second one is given by the k-th Fourier components H(Xνred)k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(X^{\ extrm{red}}_{\ u })_k$$\\end{document} induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-coefficient when X is a CR manifold with non-degenerate Levi form.

A Remark on Oka’s Lemma and a Geometric Property of Pseudoconvex Domains

A direct proof of Oka’s lemma on the relation of holomorphic convexity and the properties of the distance to the boundary function is provided. Some related problems for subharmonicity properties of this function are also studied. A new geometric property of pseudoconvex domains is described.

A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Let Ω \Omega be a strictly pseudoconvex domain in C n \mathbb {C}^n with C k + 2 C^{k+2} boundary, k ≥ 1 k \geq 1 . We construct a ∂ ¯ \overline \partial solution operator (depending on k k ) that gains 1 2 \frac 12 derivative in the Sobolev space H s , p ( Ω ) H^{s,p} (\Omega ) for any 1 > p > ∞ 1>p>\infty and s > 1 p − k s>\frac {1}{p} -k . If the domain is C ∞ C^{\infty } , then there exists a ∂ ¯ \overline \partial solution operator that gains 1 2 \frac 12 derivative in H s , p ( Ω ) H^{s,p}(\Omega ) for all s ∈ R s \in \mathbb {R} . We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of “anti-derivative operators” for distributions defined on bounded Lipschitz domains.

The Kobayashi Metric and Gromov Hyperbolicity on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

The Kobayashi Metric and Gromov Hyperbolicity on Pseudoconvex Domains of Finite Type in $${\mathbb {C}}^2$$

On the higher order Kobayashi pseudometric

ABSTRACT We study the higher order Kobayashi pseudometric introduced by Yu. We first obtain estimates of this pseudometric in a special pseudoconvex domain in C 3 . We then study the structure of the higher order extremal discs and their connection with the standard extremal discs for the Kobayashi metric.

The commutator of the Bergman projection on strongly pseudoconvex domains with minimal smoothness

Consider a bounded, strongly pseudoconvex domain D⊂Cn with minimal smoothness (namely, the class C2) and let b be a locally integrable function on D. We characterize boundedness (resp., compactness) in Lp(D),p>1, of the commutator [b,P] of the Bergman projection P in terms of an appropriate bounded (resp. vanishing) mean oscillation requirement on b. We also establish the equivalence of such notion of BMO (resp., VMO) with other BMO and VMO spaces given in the literature. Our proofs use a dyadic analog of the Berezin transform and holomorphic integral representations going back (for smooth domains) to N. Kerzman & E. M. Stein, and E. Ligocka.

Carleson measures on convex domains with smooth boundary of finite type

AbstractFollowing M. Abate and A. Saracco's work on strongly pseudoconvex domains in , we characterize Carleson measures of in bounded convex domains in with smooth boundary of finite type. We also give examples of Carleson measures with uniformly discrete (with respect to the Kobayashi distance) sequences.

Miscellanea on PSH Functions and $$L^2$$ Methods on Pseudoconvex Domains

Miscellanea on PSH Functions and $$L^2$$ Methods on Pseudoconvex Domains

Capacities, Green function, and Bergman functions

AbstractUsing the logarithmic capacity, we give quantitative estimates of the Green function, as well as precise lower bounds of the Bergman kernel for bounded pseudoconvex domains in and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies the sharp estimate if is a bounded pseudoconvex domain with ‐boundary.

Hölder continuity of solutions of the Dirichlet problem in pseudoconvex domains

This paper aims to study the Dirichlet problem to the complex Monge–Ampère operator in pseudoconvex domains of strictly plurisubharmonic type. We prove the Hölder continuity of solutions to this problem.

Morse Index of Free Boundary Disk in Pseudoconvex Domain

In this paper, we study the Morse index for the overline{partial }-energy of a non-holomorphic disk in a strictly pseudoconvex domain in mathbb {C}^n or in a Kähler manifold with non-negative bisectional curvature. We give a proof that a overline{partial }-energy minimizing disk is holomorphic; in fact, more generally we show that a non-holomorphic critical disk for the overline{partial }-energy has Morse index at least n-1. We also extend the result to domains which satisfy the weaker k-pseudoconvexity property for kge 2.