Pseudoconvex Boundary Research Articles

On the Boundary Behaviour of the Squeezing Function near Weakly Pseudoconvex Boundary Points

On the Boundary Behaviour of the Squeezing Function near Weakly Pseudoconvex Boundary Points

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.

On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary

The \bar{\partial} cohomology groups with polynomial growth H^{r,s}_{\mathrm{p.g.}} will be studied. It will be shown that, given a complex manifold M , a locally pseudoconvex bounded domain \Omega\Subset M satisfying certain geometric boundary conditions and a holomorphic vector bundle E\to M , H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 holds for all s\geq1 if E is Nakano positive and r=\dim{M} . It will also be shown that H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 for all r and s with r+s>\dim{M} if, moreover, \operatorname{rank}E=1 . By the comparison theorem due to Deligne, Maltsiniotis (Astérisque 17 (1974), 141–160) and Sasakura (Inst. Math. Sci. 17 (1981), 371–552), it follows in particular that, for any smooth projective variety X , for any ample line bundle L\to X and for any effective divisor D on X such that [D]|_{| D|}\geq0 , the algebraic cohomology H^{s}_{\mathrm{alg}}(X\setminus | D | , \Omega_X^r(L)) vanishes if r+s>\dim{X} .

Bergman–Einstein metric on a Stein space with a strongly pseudoconvex boundary

Bergman–Einstein metric on a Stein space with a strongly pseudoconvex boundary

Strong localization of invariant metrics

A quantitative version of strong localization of the Kobayashi, Azukawa and Sibony metrics, as well as of the squeezing function, near a plurisubharmonic peak boundary point of a domain in \({\mathbb {C}}^n\) is given. As an application, the behavior of these metrics near a strictly pseudoconvex boundary point is studied. A weak localization of the three metrics and the squeezing function is also given near a plurisubharmonic antipeak boundary point.

Homogeneous almost complex manifolds and their compact quotients

This paper investigates the (non)existence of compact quotients of the homogeneous almost-complex strongly-pseudoconvex manifolds discovered and classified by Gaussier–Sukhov [Wong–Rosay theorem in almost complex manifolds, http:www.arXiv.org:math.CV/0307335 ; On the geometry of model almost complex manifolds with boundary, Math. Z. 254(3) (2006) 567–589] and Lee [Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J. 54(1) (2006) 179–205; Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math. 623 (2008) 123–160].

Estimates for the Squeezing Function Near Strictly Pseudoconvex Boundary Points with Applications

An extension of the estimates for the squeezing function of strictly pseudoconvex domains obtained recently by J. E. Forn\ae ss and E. Wold in \cite{FW1} is applied to derive a sharp boundary behaviour of invariant metrics and Bergman curvatures.

Invariant Metrics on the Complex Ellipsoid

We provide a class of geometric convex domains on which the Caratheodory–Reiffen metric, the Bergman metric, the complete Kahler–Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Euclidean ball.

On a Holomorphic Family of Stein Manifolds with Strongly Pseudoconvex Boundaries

We study the stable embedding problem for a CR family of 3-dimensional strongly pseudoconvex CR manifolds with each fiber bounding a stein manifold.

On the Kähler-Einstein metric at strictly pseudoconvex points

ABSTRACTWe prove a boundary regularity result for the complete Kähler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary points, and we deduce the asymptotic behavior of their holomorphic bisectional curvatures near such points. Finally, we prove that the same asymptotic behavior holds at boundary points at which the squeezing function tends to one.

Equidistribution theorems on strongly pseudoconvex domains

This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szeg\H{o} kernel on $X$. In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sj\ostrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline M$.

Topology of Kähler Manifolds with Weakly Pseudoconvex Boundary

We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold K has l≥2 boundary components (possibly l=∞), then it has the first Betti number at least l−1, and the Levi form of any boundary component is zero. If K has l≥1 pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of K is at least l. In either case, any boundary component has a nonvanishing first Betti number. If K has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of K is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.

Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

We prove that for a strongly pseudoconvex domain $$D\subset \mathbb {C}^n$$ , the infinitesimal Caratheodory metric $$g_C(z,v)$$ and the infinitesimal Kobayashi metric $$g_K(z,v)$$ coincide if z is sufficiently close to bD and if v is sufficiently close to being tangential to bD. Also, we show that every two close points of D sufficiently close to the boundary and whose difference is almost tangential to bD can be joined by a (unique up to reparameterization) complex geodesic of D which is also a holomorphic retract of D. The same continues to hold if D is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed; this has consequences for the behavior of the squeezing function.

The Schwarz lemma at the boundary of the symmetrized bidisc

The symmetrized bidisc G2 is defined byG2:={(z1+z2,z1z2)∈C2:|z1|<1,|z2|<1,z1,z2∈C}. It is a bounded inhomogeneous pseudoconvex domain without C1 boundary, and especially the symmetrized bidisc hasn't any strongly pseudoconvex boundary point and the boundary behavior of both Carathéodory and Kobayashi metrics over the symmetrized bidisc is hard to describe precisely. In this paper, we study the boundary Schwarz lemma for holomorphic self-mappings of the symmetrized bidisc G2, and our boundary Schwarz lemma in the paper differs greatly from the earlier related results.

Compactness theorems for sequences of pseudo-holomorphic coverings between domains in almost complex manifolds

Our aim in this paper is to characterize smooth domains (D, J) and $$(D',J')$$ in almost complex manifolds of real dimension $$2n+2$$ with a covering orbit $$\{f_k (p)\}$$ , accumulating at a strongly pseudoconvex boundary point, for some $$(J,J')$$ -holomorphic coverings $$f_k : (D,J)\rightarrow (D', J')$$ and $$p\in D$$ . It was shown that such domains are both biholomorphic to a model domain, if the source domain (D, J) admits a bounded strongly J-plurisubharmonic exhaustion function. Furthermore, if the target domain $$(D',J')$$ is strongly pseudoconvex, then both (D, J) and $$(D',J')$$ are biholomorphic to the unit ball in $${\mathbb C}^{n+1}$$ with the standard complex structure. Our results can be considered as compactness theorems for sequences of pseudo-holomorphic coverings. Lin and Wong (Rocky Mt J Math 20(1):179–197, 1990) and Ourimi (Proc AMS 128(3):831–836, 2000) generalize for relatively compact domains in almost complex manifolds.

Estimate of the squeezing function for a class of bounded domains

We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point.

Intrinsic derivative, curvature estimates and squeezing function

This survey paper consists of two folds. First of all, we recall the concept of intrinsic derivative which was introduced by Lu (1979) and the related works due to Lu in his last ten years, including the holomorphically isometric embedding into the infinite dimensional Grassmann manifold and the Bergman curvature estimates for bounded domains in ℂn. Inspired by Lu’s idea, we give the lower and upper bounds estimates for the Bergman curvatures in terms of the squeezing function—one concept originally introduced by Deng et al. (2012). Finally, we survey some recent progress on the asymptotic behaviors for Bergman curvatures near the strictly pseudoconvex boundary points and present some open problems on the squeezing functions of bounded domains in ℂn.

Tangential Cauchy–Riemann Equations on Pseudoconvex Boundaries of Finite and Infinite Type in $$\mathbb {C}^2$$ C 2

In this paper, we will present what we think is a natural extension of the finite type condition in the sense of Range on pseudoconvex domains. This type condition generalizes the notion of finite type in the original theory as well as consists many cases of infinite type in the sense of Range. Following the integral representation by Henkin, we also provide \(L^p\) and “new” Holder estimates for solutions of \(\bar{\partial }_b\)-equations on the boundaries of domains which emphasize the infinite type condition.

On thickening of holomorphic hulls and envelopes of holomorphy on Stein spaces

On a normal Stein variety [Formula: see text], we study the thickening problem, i.e. the problem whether the assumption that a compact set [Formula: see text] is contained in the interior of another compact set, [Formula: see text] implies that the same inclusion holds for their holomorphic hulls. An affirmative answer is given for [Formula: see text] with isolated quotient singularities. On any Stein space [Formula: see text] with isolated singularities, we prove thickening for those hulls which have analytic structure at the singular points, obtaining a limitation for possible counter-examples. In dimension [Formula: see text], we finally relate the holomorphic hulls to analytic extension from parts of strictly pseudoconvex boundaries.