Real Analytic Boundary Research Articles

On boundary regularity of proper holomorphic mappings

We show that a proper holomorphic mapping \(f:D\to D'\) from a domain \(D\Subset{\mathbb C}^n\) with real-analytic boundary to a domain \(D'\Subset{\mathbb C}^n\) with real-algebraic boundary extends holomorphically to a neighborhood of \(\overline D\).

On Duality in Spaces of Solutions to Elliptic Systems

Suppose that D is a bounded domain in \(\mathbb{R}^n \) (n≥2) with connected real-analytic boundary, A is an elliptic system with real-analytic coefficients in a neighborhood of the closure \(\overline D \) of D, and sol(A,D) is the space of solutions to the system Au=0 in D furnished with the standard Frechet–Schwartz topology. Then the dual of sol(A,D) represents the space sol(A,\(\overline D \)) of solutions to the system Au=0 in a neighborhood of \(\overline D \) furnished with the standard inductive limit topology over some decreasing net of neighborhoods of \(\overline D \). The corresponding pairing is generated by the inner product in the Lebesgue space L2(D).

Analytic singularities of the Bergman kernel for tubes

Let Ω⊂ℝn be a bounded, convex, and open set with real analytic boundary. Let TΩ⊂ℂn be the tube with base Ω, and let $\mathcal{B}$ be the Bergman kernel of TΩ. If Ω is strongly convex, then $\mathcal{B}$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation we relate the off-diagonal points where analyticity fails to the characteristic lines. These lines are contained in the boundary of TΩ, and they are projections of the Treves curves. These curves are symplectic invariants that are determined by the CR (Cauchy-Riemann) structure of the boundary of TΩ. Note that Treves curves exist only when Ω has at least one weakly convex boundary point.

A Note on a Theorem of H. Cartan

We prove that if D ⊃ C is a bounded domain with real analytic boundary, and D is either pseudoconvex or it satisfies condition (R), then the compact open topology in Aut(D), the group of holomorphic automorphisms of D is the topology of uniform convergence on D.

On the Scattering Length Spectrum for Real Analytic Obstacles

It follows trivially from old results of Majda and Lax–Phillips that connected obstacles K with real analytic boundary in Rn are uniquely determined by their scattering length spectrum. In this paper we prove a similar result in the general case (i.e. K may be disconnected) imposing some non-degeneracy conditions on K and assuming that its trapping set does not topologically divide S*(C), where C is a sphere containing K. It is shown that the conditions imposed on K are fulfilled, for instance, when K is a finite disjoint union of strictly convex bodies.

A Forelli–Rudin Construction and Asymptotics of Weighted Bergman Kernels

Let Ω be a pseudoconvex domain in CN with smooth boundary, −φ, −ψ two smooth defining functions for Ω={φ>0} such that −logψ, −logφ are plurisubharmonic, z∈Ω a point at which −logφ is strictly plurisubharmonic, and M⩾0 an integer. We show that as k→∞, the Bergman kernels with respect to the weights φkψM have the asymptotic expansionKφkψM(z, z)=kNπNφ(z)kψ(z)M∑j=0∞bj(z)k−j,b0=det−∂2logφ∂zj∂zk.For Ω strongly pseudoconvex with real-analytic boundary, φ, ψ real analytic and −logφ, −logψ strictly plurisubharmonic on Ω, we obtain also the analogous result for KφkψM(x, y) for (x, y) near the diagonal and discuss consequences for the asymptotics of the Berezin transform and for the Berezin quantization. The proofs rely on Fefferman's expansion for the Bergman kernel in a certain Forelli–Rudin type domain over Ω; as another application, they also yield a generalization of the cited Fefferman's expansion to a class of weighted Bergman kernels.

Factorization of proper holomorphic mappings through Thullen domains

In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$ satifying: (a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and (b) it is simply connected and has a real analytic boundary. According to [Barletta-Bedford, Indiana U. Math. J, 39(1985), 315-338], the strong pseudconvexity of $B^2$ alone yields that such a domain is at the boundary points that are at the same time a smooth point of the branch locus $Z_{df} = \{\det(J_{\bf C} f) = 0\}$. (Notice that [Diederich-Fornaess, Math. Ann., 282 (1988), 681-700] implies that $f$ as well as $Z_{df}$ extends holomorphically across the boundaries.) Our main contribution in this paper is that we have discovered a stronger rigidity (both local and global) in case the target domain is the unit ball. The main results are: THEOREM (Local Rigidity): Let $(M,o)$ be a real analytic normalized weakly spherical pointed CR hypersurface in ${\bf C}^2$ of order $k_0 > 1$. Let $(\Sigma, o)$ be the pointed Siegel hypersurface given by the defining equation $Re w - |z|^2 = 0$. If there is a holomorphic mapping $F:(M,o) \to (\Sigma,o)$ for which $o$ is a regular branch point, then (1) $(M,o)$ is defined by the equation $Re w - |z|^{2k_0} = 0$, and (2) $F(z,w)$ is equivalent to $(z,w) \mapsto (z^{k_0},w)$ up to a composition with elements in $Aut (M,o)$ and $Aut (\Sigma,o)$. THEOREM (Global Rigidity): Let $D$ and $f:D \to B^2$ be as above, and let $f$ be generically $m$-to-1. Assume that its branch locus $Z_{df}$ admits an analytic component $V$ with the following properties: (1) $f$ is locally a $m$-to-1 branched covering with branch locus $V$ at every point of $V \cap \partial D$; (2) $V \cap \partial D$ is connected and contains no singular point of the variety $Z_{df}$. Then $D$ is biholomorphic to $E_m = \{|z|^{2m} + |w|^2 < 1\}$.

Failure of global regularity of ∂b on a convex domain with only one flat point

In this paper we exhibit a bounded domain in C 2 with real analytic boundary which is strictly convex except at one point and for which the @b operator is not analytic hypoelliptic modulo its kernel. The importance of such an example is twofold: First it shows that the theorem of Boas and Straube on global C 1 regularity for @b on convex domains cannot be extended to the analytic case; secondly it is the rst example of non analytic hypoellipticity of @b on a domain with isolated weakly pseudoconvex points in the boundary.

On reflection principle in several complex variables

Let J be a holomorphic mapping where is a domain with real analytic boundary ∂Ω such that . Suppose J maps ∂Ω into an algebraically normal real hypersurface ∑ By closely following the argument of Boauendi and Rotschild |2. 4| we prove the following results. Theorem 1: If is rigid and J is nondegenerate at point . then J extends holomorphically to a full neighbourhood of ρ. Theorem 2: If ∂Ω and ∑ are of infinite type and J not completely degenerate, then J extends holomorphically to a full neighborhood of ρ

Analytic regularity for the Bergman kernel

Let Ω⊂ℝ 2 be a bounded, convex and open set with real analytic boundary. Let T Ω ⊂ℂ 2 be the tube with base Ω, and let ℬ be the Bergman kernel of T Ω . If Ω is strongly convex, then ℬ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of T Ω . Note that Trèves curves exist only when Ω has at least one weakly convex boundary point.

On an 𝑛-manifold in 𝐂ⁿ near an elliptic complex tangent

In this paper, we study the local biholomorphic property of a real n n -manifold M ⊂ C n M\subset \mathbf C^n near an elliptic complex tangent point p ∈ M p\in M . In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy M ~ \widetilde {M} of M M near p p , first considered in a paper of Bishop. When M M is a C ∞ C^{\infty } -smooth submanifold, using a result established by Kenig-Webster, we show that near p p , M ~ \widetilde {M} is a smooth Levi-flat ( n + 1 ) (n+1) -manifold with a neighborhood of p p in M M as part of its C ∞ C^{\infty } boundary. Moreover, near p p , M ~ \widetilde {M} is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to M M . This result was proved in the previous work of Kenig-Webster when n = 2 n=2 . When M M is real analytic, we show that M ~ \widetilde {M} is real analytic with a neighborhood of p p in M M as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When n = 2 n=2 or when n &gt; 2 n&gt;2 but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author’s joint work with Krantz.

Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains

In his paper of 1907 [Po07], Poincare showed, among other things, that any local non-constant holomorphic map f from an open piece of the sphere ∂B into ∂B extends to a global biholomorphic map between the ball B. This result was generalized to B by Tanaka [Ta62] and was further established in a much more general setting by Alexander [A74]. Since then, the mixed equivalence problem was naturally formulated (see for instance [Wel82], [Be90]): How far can a local equivalence map between the boundaries of two nice domains be biholomorphically extended ? A significant contribution along the lines of this direction was made by Webster: In [We77], he not only extended Poincare’s theorem to real ellipsoids, but also did a pioneer work by showing a type of Chow’s theorem for local maps between algebraic domains. In particular, he showed that any local equivalence map between two smooth algebraic domains extends as a branched algebraic map. Webster’s algebraicity theorem was established in more general situations in the recent deep work of Baouendi, Ebenfelt and Rothschild (see [BR95], [BER96]), and in the first author’s work ([Hu94], [Hu94T]). In [CJ96], Chern and the second author showed a related result: Any local map f , which maps a piece of ∂D into ∂B, extends along any path γ ⊂ D as a locally bimeromorphic map, where D is a bounded domain with real analytic spherical boundary. The purpose of this paper is to study the global holomorphic extension of a local map between algebraic domains, i.e, domains whose boundaries are locally defined by real polynomials. Our first result is the following theorem:

Peak curves in weakly pseudoconvex boundaries in C2

LetD be a pseudoconvex domain with real analytic boundary in C2. A subsetE of ∂D is a local peak set for Open image in new window if for everyp ∈ ∂D, there exist a neighborhoodU ofp and a holomorphic functionf onU such thatf = 1 onE∩U and ¦f¦ < 1 on\(\bar D \cap U\backslash E\). We give conditions for the existence of real analytic LPι curves in ∂D through a point of finite type.

Classical area minimizing surfaces with real-analytic boundaries

Classical area minimizing surfaces with real-analytic boundaries

On the uniform extendibility of proper holomorphic mappings

This paper considers the problem oi holomorphic extension of compact families of holomorphic proper mappings between bounded pseudoconvex domains with real analytic boundaries. Some results on the uniform extension are established We prove a general version of Vitushkin's theorem.

Examples of Domains with Non-Compact Automorphism Groups

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any Reinhardt domain. We give an analogous example in dimension two, but the domain fails to be smooth at one boundary point---indeed it is not in any Lipschitz class at the exceptional point.

Harmonic duality, hyperfunctions and removable singularities

This article is devoted to the study of the boundary behavior of holomorphic functions on strictly pseudoconvex domains with real-analytic boundary in and to the theory of removable sets for such boundary values. The work depends on some preliminary results on the boundary values of harmonic functions on domains in with real-analytic boundaries and on a new result concerning the dual space of all such functions on a given domain.

Nodal volumes for eigenfunctions of analytic regular elliptic problems

We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions of a regular elliptic problem on a bounded domain Ω ⊆ ℝn with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary.

Nonvanishing of the differential of holomorphic mappings at boundary points

In this paper we prove a general result of the ``Hopf lemma'' type for CR mappings, with nonidentically vanishing Jacobians, between real hypersurfaces in C^n with smooth or real analytic boundaries. Applications of this result to finiteness and holomorphic extendibility of such mappings are also given. The novelty here is that we make no assumption on the nonflatness of the mapping or its Jacobian, nor do we assume that the hypersurfaces are pseudoconvex or minimally convex.

On the Stability of the Spectrum in the Pompeiu Problem

Let Ω be a Jordan domain in the complex plane with smooth boundary. We call the Pompeiu spectrum σ( Ω) the set of all λ such that there exists a nontrivial solution of overdetermined Dirichlet-Neumann boundary-value problem. [formula] (ν is the normal vector to the boundary ∂ Ω). Let Ω t , t ∈ [0, T) be a family of Jordan domains in the complex plane with real-analytic boundaries. Suppose that Ω t analytically depends on the parameter t and Ω 0 = { z ∈ C : | z| ≤ 1}. It is proved that if there exists a real-analytic function λ( t), such that λ( t) ∈ σ( Ω t ), t ∈ [0, T), then all domains Ω t are discs.