Strictly Pseudoconvex Research Articles

A Constraint on Chern Classes of Strictly Pseudoconvex CR Manifolds

This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal'' through some examples.

A remark on boundary estimates on unbounded Z(q) domains in

The goal of this note is to explore the relationship between the Folland–Kohn basic estimate and the Z(q)-condition. In particular, on unbounded domains, we prove that the Folland–Kohn basic estimate is equivalent to a uniform Z(q) condition. As a corollary, we observe that despite the Siegel upper half space being strictly pseudoconvex and biholomorphic to the unit ball, it fails to satisfy uniform strict pseudoconvexity and hence the Folland–Kohn basic estimate fails.

On Traces in Hardy Type Analytic Spaces in Bounded Strictly Pseudoconvex Domains and in Tubular Domains over Symmetric Cones

On Traces in Hardy Type Analytic Spaces in Bounded Strictly Pseudoconvex Domains and in Tubular Domains over Symmetric Cones

Affine-homogeneous real hypersurfaces of tube type in ℂ3

In the present article, the scheme is proposed for constructing a complete description of affine-homogeneous strictly pseudo-convex (SPC) real hypersurfaces of 3-dimensional complex space. The scheme is illustrated by the example of the class of tube-type hypersurfaces that generalizes the known class of tubular manifolds. The estimate is obtained for the dimension of affine transformations group acting on such a surface. It is shown that all homogeneous surfaces of tube type with rich (7-dimensional and 6-dimensional) symmetry groups are affinely equivalent to tubular surfaces (tubes). It is proved that every affine-homogeneous SPC-tube of the space ℂ3 is affinely equivalent to a tube with homogeneous base. Examples are given of affine-homogeneous surfaces of tubular type with 5-dimensional groups that could not be reduced to tubes by means of affine transformations.

An example of a two-parameter family of affine homogeneous real hypersurfaces in ℂ3

This paper studies real hypersurfaces in multidimensional complex spaces homogeneous with respect to holomorphic transformations. In the case of spaces of two complex variables, this problem was completely solved in explicit form as early as in 1932 by Cartan [1]. But even in the problem of describing holomorphic homogeneous real hypersurfaces in complex 3-spaces, there are many questions that remain unanswered so far. Therefore, it is natural to first study the simpler problem of affine homogeneous hypersurfaces in complex space C3. We emphasize that the family of homogeneous real hypersurfaces in C3 is very large (see [2], [3]). For this reason, an important part of the problems stated above is not only finding new families of homogeneous manifolds but also determining their position in some (desirably, complete) classification. Results of the papers [2] and [3] mentioned above show that both the holomorphic and the affine homogeneity of hypersurfaces in multidimensional spaces can be studied within the framework of a unified approach related to local canonical equations of the manifolds under consideration. For example, in [3], by analogy with the well-known (holomorphic)Moser normal form [4], the notion of an affine canonical equation of a real hypersurface in C3 was suggested. According to [3], in a neighborhood of any point, the equation of a strictly pseudoconvex (SPC) real-analytic (not necessarily homogeneous) hypersurface in this space can be reduced by affine transformations to the form

On Strict Pseudoconvexity

The present paper provides first and second-order characterizations of a radially lower semicontinuous strictly pseudoconvex function ƒ : X → ℝ defined on a convex set X in the real Euclidean space ℝ n in terms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [J. Econom. Theory 25: 397–420, 1981]. A new implication appears in our conditions. Connections with other classes of functions are also derived.

Dynamics of Local Automorphisms of Embedded CR-Manifolds

A classical theorem due to Wong [1] states that the ball is the unique strictly pseudoconvex bounded domain having a noncompact automorphism group. Rosay [2] extended this theorem to bounded domains such that an orbit accumulates at a strictly pseudoconvex domain at the boundary. Later, Efimov [3] got rid of the boundedness assumption. Schoen [4] and, independently, Spiro [5] proved the CR-version of this result, where the strict pseudoconvexity of the accumulation point of an orbit also plays a crucial role. In the present note, we suggest two local CR-versions of this result for hypersurfaces. These versions manifest different behavior depending on whether the Levi form is definite or not. Let M be a real-analytic Levi-nondegenerate hypersurface in some neighborhood U of the origin in C , n ≥ 1 . Denote by Aut0 M the group of local diffeomorphisms of M , that is, the group of germs of biholomorphisms taking M locally into itself and preserving the origin. A point q = 0 (treated as its own radius-vector −→ 0q) is said to be contractible if there is a sequence of local automorphisms Φn ∈ Aut0 M such that

Weak pseudoconcavity and the maximum modulus principle

In this paper we focus on the maximum modulus principle and weak unique continuation for CR functions on an abstract almost CR manifold M. It is known that some assumption must be made on M in order to have either of these: it suffices to consider the standard CR structure on the sphere S3 in C2 to see that the maximum modulus principle is not valid in the presence of strict pseudoconvexity. For weak unique continuation, Rosay [R] has shown by an example that there is a strictly pseudoconvex CR structure on R3, which is a perturbation of the aforementioned standard CR structure on S3, such that there exists a smooth CR function u, u ≡ 0, with u ≡ 0 on a nonempty open set. However positive results were obtained in [DCN] under the assumption of pseudoconcavity and in [HN] under the assumption of essential pseudoconcavity (and also the finite kind for the maximum modulus principle). Here we investigate these matters under the assumption of weak pseudoconcavity on M, which is a more general notion than that of essential pseudoconcavity, insofar as it drops the minimality (and the finite kind) hypothesis on M. We obtain sharp results involving propagation along Sussmann leaves. The core of our argument is that on a weakly pseudoconcave M the square of the modulus of a CR function is subharmonic with respect to a degenerate-elliptic operator P on M. We employ a maximum principle for real-valued functions which is in the spirit of [Hf], [Ni], [B], [H]. In order to understand our motivation in considering the weak pseudoconcavity condition on M, the reader is referred to the examples in [HN].

Microlocal analyticity for the canonical solution to on Strictly Pseudoconvex CR Mainfolds of Real Dimension Three

We show that {bar {partial_derivative}}{sub b}* is analytic hypoelliptic on one of its characteristic directions and that {open_square}{sub b} = {bar {partial_derivative}}{sub b}{bar {partial_derivative}}{sub b}* is in the opposite one on strongly pseudoconvex manifolds of dimension 2 on which {bar {partial_derivative}}{sub b} has closed range. It follows that the canonical solution to {bar {partial_derivative}}{sub b}u = f is analytical microlocally where f is. This gives an elementary proof of Geller`s theorem on the Szegoe projector, and, with a fully mirolocal version of the necessary and sufficient condition for hypoellipticity with loss of one derivative, unifies the results of Trepreau in Treves.

On three-dimensional Cauchy-Riemann manifolds

of the almost complex tensor of C2 (multiplication by i ). The collection of these subspaces Hp M forms a bundle HM c TM called the horizontal or contact bundle. (Indeed, strict pseudoconvexity of M implies that the plane field {HpM} is nondegenerate, i.e. defines a contact structure.) The almost complex tensor of C2 then restricts to a bundle endomorphism J: HM -HM such that J2 = -id. The bundle HM c TM together with this endomorphism J defines the CR structure of M. Slightly more generally, let D = D U M be a four dimensional compact manifold with boundary M, interior D, endowed with a smooth almost complex tensor which is integrable on D. If D is strictly pseudoconvex-i.e., in a neighborhood of any p E M there is a smooth strictly plurisubharmonic function u, negative on D, 0 on M, but du $ 0 on M-then M inherits a strictly pseudoconvex CR structure from D. In this case we shall say that M bounds a strictly pseudoconvex surface. This leads us to the abstract definition. A strictly pseudoconvex (threedimensional) CR manifold is a compact manifold M (without boundary), dim M = 3, endowed with a contact structure HM = {HpM: p E M} c TM and an endomorphism J of HM such that J2 = -id. (Throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects. Thus M, HM, J are assumed to be smooth in this sense.) Furthermore, a differentiable function f: M -? C is a CR function if for any p E M the restriction df IHM: HpM -Tf(p,)C

Extension of the Complex Structure from Stein Manifolds with Strictly Pseudoconvex Boundary

Mathematische NachrichtenVolume 128, Issue 1 p. 57-64 Article Extension of the Complex Structure from Stein Manifolds with Strictly Pseudoconvex Boundary Dieter Heunemann, Dieter Heunemann Akademie der Wissenschaften der DDR, Karl-Weierstraß-Institut für Mathematik, DDR-1086 Berlin, Mohrenstraße 39Search for more papers by this author Dieter Heunemann, Dieter Heunemann Akademie der Wissenschaften der DDR, Karl-Weierstraß-Institut für Mathematik, DDR-1086 Berlin, Mohrenstraße 39Search for more papers by this author First published: 1986 https://doi.org/10.1002/mana.19861280105Citations: 4AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume128, Issue11986Pages 57-64 RelatedInformation

Theorem B for Stein Manifolds with Strictly Pseudoconvex Boundary

Mathematische NachrichtenVolume 128, Issue 1 p. 87-101 Article Theorem B for Stein Manifolds with Strictly Pseudoconvex Boundary Dieter Heunemann, Dieter Heunemann Akademie der Wissenschaften der DDR, Karl-Weierstraß-Institut für Mathematik, DDR-1086 Berlin, Mohrenstraße 39Search for more papers by this author Dieter Heunemann, Dieter Heunemann Akademie der Wissenschaften der DDR, Karl-Weierstraß-Institut für Mathematik, DDR-1086 Berlin, Mohrenstraße 39Search for more papers by this author First published: 1986 https://doi.org/10.1002/mana.19861280108Citations: 12AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume128, Issue11986Pages 87-101 RelatedInformation

Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of ${\bf C}\sp{n}$

The relationship between the Levi geometry of a submanifold of C' and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of Cn, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in Cn. This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of Cn. In fact, we show that if S is a real analytic, generic, submanifold of C' (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C. 0. Introduction. Recently, there has been increasing interest in the relationship of geometric concepts, such as the Levi form of a hypersurface and the local solvability of the tangential Cauchy-Riemann equations. For example, Henkin [H] has shown that on a strictly pseudoconvex hypersurface, one can locally solve the tangential Cauchy-Riemann equations (in most bidegrees). His approach involved explicit integral kernels. In this paper we start such a program for higher codimension. We relate such concepts as the Levi form and the Levi algebra to the local solvability of the tangential Cauchy-Riemann equations on a real analytic, codimension two submanifold of Cn. The geometric conditions we impose are in terms of strict pseudoconvexity of the Levi form (Definition 2.3) and the excess dimension of the Levi algebra (Definition 2.2). We represent the local solution to the tangential Cauchy-Riemann equations in terms of explicit kernels. We show (Theorem 5.1) that on a generic real analytic, strictly pseudoconvex, codimension two submanifold S of Cn with excess dimension of S equal to one, we can locally solve the tangential Cauchy-Riemann equations in bidegrees (p, q) where 1 s q < n 3. Although our emphasis is on codimension two, we indicate an inductive process for local solvability on higher codimension submanifolds of Cn. From the kernel approach we also get, for free, a local result (Theorem 5.15) about CR-functions on a codimension two submanifold, S, satisfying the hypothesis of Theorem 5.1. Locally, we carefully construct an open set V in C n with V n s F 0 such that a CR-function on V n s is the boundary value jump across S of a Received by the editors January 24, 1980 and, in revised form, February 9, 1981. 1980 Mathematics Subject Classification. Primary 35N15, 32F99. (DI 982 American Mathematical Society 0002-9939/82/00001064/$06.75

Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions

The complex analysis of strictly pseudoconvex domains in IE" is rather well known, especially since the boundary regularity properties of solutions of the inhomogeneous Cauchy-Riemann equations 8c~=fl on such domains have been described more and more precisely and important consequences from this have been derived (for a survey of the results in this direction known until 1972 (see [8]), more recent results are for instance contained in [15, 23]). In contrast to this situation, many fundamental questions with respect to the most "natural" domains of complex analysis, namely arbitrary pseudoconvex domains in ~ ' , are still open. What is the reason for this discrepancy? One possible answer to this question can be seen in the following observation. In almost all important proofs concerning the analysis of strictly pseudoconvex domains, at least one out of three elementary properties of these domains play a fundamental role. They are: 1) Strict pseudoconvexity is stable with respect to small C2-perturbations. 2) A strictly pseudoconvex domain is locally biholomorphically equivalent to strictly convex domains. 3) If ( 2 c c ~ " is strictly pseudoconvex with a smooth C2-boundary, then there is a neighborhood U of ~ and a strictly plurisubharmonic C 2 function p on U such that dp(p):t=O for peb~2 and