Pseudoconvex Hypersurfaces Research Articles

The sharp upper bounds for the first positive eigenvalue of the Kohn–Laplacian on compact strictly pseudoconvex hypersurfaces

We give sharp and explicit upper bounds for the first positive eigenvalue $\lambda_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $\lambda_1(\Box_b)$ has a unique maximum value at the CR sphere.

Convergence of formal CR mappings into strongly pseudoconvex Cauchy\u2013Riemann manifolds

It is shown that any formal holomorphic mapping sending a real-analytic generic submanifold Msubset mathbb {C}^N of finite type into a real-analytic strongly pseudoconvex CR submanifold M'subset mathbb {C}^{N'} is necessarily convergent. As a consequence, we obtain a positive answer to the long-standing open question of whether formal holomorphic maps sending real-analytic strongly pseudoconvex hypersurfaces into each other are convergent.

Umbilical points on three dimensional strictly pseudoconvex CR manifolds I: manifolds with U(1)-action

The question of existence of umbilical points, in the CR sense, on compact, three dimensional, strictly pseudoconvex CR manifolds was raised in the seminal paper by S.-S. Chern and J. K. Moser in 1974. In the present paper, we consider compact, three dimensional, strictly pseudoconvex CR manifolds that possess a free, transverse action by the circle group $U(1)$. We show that every such CR manifold $M$ has at least one orbit of umbilical points, {\it provided} that the Riemann surface $X:=M/U(1)$ is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in $\mathbb C^2$ has at least one circle of umbilical points. The existence of umbilical points in the case where $X$ is a torus is left open in general, but it is shown that if such an $M$ has additional symmetries, in a certain sense, then it must possess umbilical points as well.

Affine-homogeneous surfaces of type (0, 0) in the space ℂ3

The problem of describing affine homogeneous real hypersurfaces of the complex space C3 was posed in [1]. Such an affine setting is closely related to the problem of classifying holomorphically homogeneous submanifolds of multidimensional complex spaces (see [2]–[4]). A scheme for the study of the affine homogeneity property based on the use of the canonical equations of the manifolds under consideration was proposed in [5]–[7]. The scheme envisages a separate study of several (for example, seven) types of strictly pseudoconvex hypersurfaces depending on the second-order Taylor coefficients of their canonical equations. This scheme was used in [6]–[8] to give a complete description of two of the seven cases mentioned above. Below we describe the third possible type.

Stationary Discs for Smooth Hypersurfaces of Finite Type and Finite Jet Determination

We construct a finitely dimensional manifold of invariant holomorphic discs attached to a certain class of smooth pseudoconvex hypersurfaces of finite type, generalizing the notion of stationary discs. The discs we construct are determined by a finite jet at a given boundary point and their centers fill an open set. As a consequence, we obtain finite jet determination of CR diffeomorphisms between smooth hypersurfaces of this class.

On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly psedoconvex hypersurface $M^{2n+1}\subset\bC^{n+1}$ into the sphere $\bS^{2N+1}\subset \bC^{N+1}$ is rigid, i.e. any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere $\bS^{2N+1}$, {\it provided} that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e. we consider local CR embeddings $f\colon M^5\to \bS^7$. We show that there are at most two different local embeddings, up to composition with an automorphism of $\bS^7$. We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces $M^5$ in terms of their CR curvatures such that rigidity holds for local CR embeddings $f\colon M^5\to \bS^7$.

A characterization of hyperbolic Kato surfaces

We give a characterization of hyperbolic Kato surfaces in terms of the existence of an automorphic Green function on a cyclic covering. This is achieved by analysing a naturally dened Levi-at foliation, and by perturbing certain Levi-at leaves to strictly pseudoconvex hypersurfaces. 2010 Mathematics Subject Classication: Kato surfaces, Levi-at foliations,

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M⊂Cn+1 into a sphere S in a higher dimensional complex space CN+1. The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is <n (here, n is the CR dimension of M), then f(M) is contained in a complex plane of dimension n+d+1. The converse of this statement is also true, as is easy to see. When the total rank d exceeds n, it is no longer true, in general, that f(M) is contained in a complex plane of dimension n+d+1, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n, then partial rigidity may still persist, but there is a “defect” k that arises from the ranks exceeding n such that f(M) is only contained in a complex plane of dimension n+d+k+1. Moreover, this defect occurs in general, as is illustrated by examples.

Non-embeddable real algebraic hypersurfaces

We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.

Nonsmooth hypersurfaces with smooth Levi curvature

In R3, Lipschitz continuous viscosity solutions to the K-prescribed Levi curvature equation are smooth and strictly pseudoconvex if K is smooth and strictly positive; see [5]. We show here that in R2n+1, n>1, a similar result does not hold; that is, we prove the existence in Cn+1, n>1, of nonsmooth pseudoconvex hypersurfaces with smooth Levi curvature.

The Beltrami equations for quasiconformal mappings on strongly pseudoconvex hypersurfaces

We show that quasiconformal mappings on strongly pseudoconvex hypersurfaces satisfy a system of Beltrami equations. In particular, the 1-quasiconformal mappings on these surfaces are CR or anti-CR mappings. Furthermore, if these surfaces are also real-analytic and nonspherical, then 1-quasiconformal mappings on them, which fix a point, can be linearized.

Pseudohermitian invariants and classification of CR mappings in generalized ellipsoids

Given a strictly pseudoconvex hypersurface M ‰ C n¯1 , we discuss the problem of classifying all local CR diffeomorphisms between open subsets N , N 0 ‰ M . Our method exploits the Tanaka‐Webster pseudohermitian invariants of a contact form# on M , their transformation formulae, and the Chern‐Moser invariants. Our main application concerns a class of generalized ellipsoids where we classify all local CR mappings.

Effective Cartan–Tanaka connections for [formula omitted]-smooth strongly pseudoconvex hypersurfaces [formula omitted

Explicit Cartan–Tanaka curvatures, the vanishing of which characterizes sphericity, are provided in terms of the 6-th order jet of a graphing function for a C 6 strongly pseudoconvex hypersurface M 3 ⊂ C 2 .

On convex to pseudoconvex mappings

In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of R 3 \mathbb {R}^{3} which sends convex sets to convex ones is affine. It can be remarked also that a C 2 \mathcal {C}^{2} -diffeomorphism F : U → U ′ F:U\to U^{’} between two domains in C n \mathbb {C}^{n} , n ≥ 2 n\ge 2 , which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic. In this paper we are interested in the self-mappings of C n \mathbb {C}^{n} which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: A C 2 \mathcal {C}^{2} -diffeomorphism F : U ′ → U F:U’\to U (where U ′ , U ⊂ C n U’, U\subset \mathbb {C}^{n} are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map Φ := F − 1 \Phi := F^{-1} is weakly pluriharmonic, i.e., if it satisfies some nice second order PDE very close to ∂ ∂ ¯ Φ = 0 \partial \bar {\partial }\Phi = 0 . In fact all pluriharmonic Φ \Phi ’s do satisfy this equation, but there are also other solutions.

On the Cauchy problem for the $\bar\partial$ operator

We present new results concerning the solvability, or lack of thereof, in the Cauchy problem for the $\bar\partial$ operator with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.

Generalized models and local invariants of Kohn–Nirenberg domains

The main obstruction for constructing holomorphic reproducing kernels of Cauchy type on weakly pseudoconvex domains is the Kohn–Nirenberg phenomenon, i.e. nonexistence of supporting functions and local nonconvexifiability. In this paper we give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models.

Subelliptic SpinℂDirac operators, I

Let X be a compact Kahler manifold with strictly pseudoconvex bound- ary, Y. In this setting, the SpinC Dirac operator is canonically identified with ¯ ∂ + ¯ ∂ ∗ : C ∞ (X ;Λ 0,e ) →C ∞ (X ;Λ 0,o ). We consider modifications of the classi- cal ¯ ∂-Neumann conditions that define Fredholm problems for the SpinC Dirac operator. In Part 2, (7), we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a SpinC Dirac operator with a subellip- tic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface, then we obtain formulae for the holomorphic Euler characteristic of X as sums of indices of SpinC Dirac operators on the components. This is a subelliptic analogue of Bojarski's formula in the elliptic case.

On the Curvature Groups of a CR Manifold

We show that any contact form whose Fefferman metric admits a nonzero parallel vector field is pseudo-Einstein of constant pseudohermitian scalar curvature. As an application we compute the curvature groups of the Fefferman space C(M) of a strictly pseudoconvex real hypersurface $$M\subset \mathbb{C}^{n+1}.$$

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.

TOTAL REALITY OF CONORMAL BUNDLES OF HYPERSURFACES IN ALMOST COMPLEX MANIFOLDS

A generalization to the almost complex setting of a well-known result by Webster is given. Namely, we prove that if Γ is a strongly pseudoconvex hypersurface in an almost complex manifold (M, J), then the conormal bundle of Γ is a totally real submanifold of (T* M, 𝕁), where 𝕁 is the lifted almost complex structure on T* M defined by Ishihara and Yano.