Analytic Discs Research Articles

On the Extension of Bounded Holomorphic Maps from Gleason Parts of the Maximal Ideal Space of

Abstract Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$ . In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$ . The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$ . The proofs utilize our recent advances on $\bar \partial $ -equations on quasi-interpolating sets and Runge-type approximations.

Testing families of analytic discs in the unit ball of ℂ2

Testing families of analytic discs in the unit ball of ℂ2

Complete norm-preserving extensions of holomorphic functions

We show that for every connected analytic subvariety V there is a pseudoconvex set Ω such that every bounded matrix-valued holomorphic function on V extends isometrically to Ω. We prove that if V is two analytic discs intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to Ω, then so does every matrix-valued function. In the special case that Ω is the symmetrized bidisc, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.

Cartan uniqueness theorem on nonopen sets

Cartan’s uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that V is a locally connected and locally closed subset of $${\mathbb {C}}^n$$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then, restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan’s uniqueness theorem for such sets. When V is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties, the only automorphisms, CR or infinitesimal, are linear.

Stability of the hull(s) of an n-sphere in [formula omitted

We study the (global) Bishop problem for small perturbations of Sn — the unit sphere of C×Rn−1 — in Cn. We show that if S⊂Cn is a sufficiently-small perturbation of Sn (in the C3-norm), then S bounds an (n+1)-dimensional ball M⊂Cn that is foliated by analytic disks attached to S. Furthermore, if S is either smooth or real analytic, then so is M (up to its boundary). Finally, if S is real analytic (and satisfies a mild condition), then M is both the envelope of holomorphy and the polynomially convex hull of S. This generalizes the previously known case of n=2 (CR singularities are isolated) to higher dimensions (CR singularities are nonisolated).

Compactness of composition operators on the Bergman spaces of convex domains and analytic discs

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in ℂn with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.

Polynomially convex embeddings of odd-dimensional closed manifolds

Abstract It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

Automorphisms of $\mathbb C^k$ with an invariant non-recurrent attracting Fatou component biholomorphic to $\mathbb C\times (\mathbb C^\ast)^{k-1}$

We prove the existence of automorphisms of \mathbb C^k , k \geq 2 , having an invariant, non-recurrent Fatou component biholomorphic to \mathbb C \times (\mathbb C^\ast)^{k-1} which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. As a corollary, we obtain a Runge copy of \mathbb C \times (\mathbb C^\ast)^{k-1} in \mathbb C^k . The constructed Fatou component also avoids k analytic discs intersecting transversally at the fixed point.

Polynomial hulls of arcs and curves

It is shown that there exist arcs and simple closed curves in C 3 \mathbb {C}^3 with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded, connected Runge domain of holomorphy in C N \mathbb {C}^N ( N ≥ 2 N \geq 2 ) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through compact, totally disconnected sets.

Nondefective Stationary Discs and 2-Jet Determination in Higher Codimension

We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of $${\mathbb C}^{N}$$ of class $${\mathcal {C}}^{4}$$ .

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

AbstractWe strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

Polynomially convex embeddings of even-dimensional compact manifolds

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth generators for the Banach algebra $\mathcal C(M)$, the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when $n=3k-1$, $k>1$, despite the presence of complex tangencies, thus lowering the known bound for the optimal $n$ in these (related but inequivalent) questions.

No topological condition implies equality of polynomial and rational hulls

It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any uncountable, compact subset K K of a Euclidean space, there exists a set X X , in some C N \mathbb {C}^N , that is homeomorphic to K K and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in C 3 \mathbb {C}^3 constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for m m -dimensional manifolds ( m ≥ 2 m\geq 2 ) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.

Spaces with polynomial hulls that contain no analytic discs

Extensions of the notions of polynomial and rational hull are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set in $${\mathbb {C}}^3$$ with a nontrivial polynomial hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and simple closed curves in $${\mathbb {C}}^4$$ with nontrivial polynomial hulls that contain no analytic discs. This answers a question raised by Bercovici in 2014 and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace X of $${\mathbb {C}}^N$$ , for some N, in such a way as to have a nontrivial polynomial hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, X can be chosen so as to have the stronger property that P(X) has a dense set of invertible elements.

Hankel operators on the Bergman spaces of Reinhardt domains and foliations of analytic disks

Let $$\Omega \subset \mathbb {C}^2$$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $$\Omega $$ and obtain a compactness result for Hankel operators on the Bergman space of $$\Omega $$.

Derivations on the algebra of Rajchman measures

For a locally compact Abelian group G, the algebra of Rajchman measures, denoted by $${M}_{0}(G)$$ , is the set of all bounded regular Borel measures on G Fourier transform of which vanish at infinity. In this paper, we investigate the spectral structure of the algebra of Rajchman measures, and illustrate aspects of the residual analytic structure of its maximal ideal space. In particular, we show that $${M}_{0}(G)$$ has a nonzero continuous point derivation, whenever G is a nondiscrete locally compact Abelian group. We then give the definition of the Rajchman algebra for a general (not necessarily Abelian) locally compact group, and prove that for a noncompact connected SIN group, the Rajchman algebra admits a nonzero continuous point derivation. Moreover, we discuss the analytic behavior of the spectrum of $${M}_{0}(G)$$ . Namely, we show that for every nondiscrete metrizable locally compact Abelian group G, the maximal ideal space of $${M}_{0}(G)$$ contains analytic disks.

Riemann-Hilbert problems with constraints

This paper is devoted to Riemann-Hilbert problems with constraints. We obtain results characterizing the existence of solutions as well as the dimension of the solution space in terms of certain indices. As an application, we show how such results may be used to construct analytic discs attached to singular manifolds.

Analytic discs and uniform algebras generated by real-analytic functions

Analytic discs and uniform algebras generated by real-analytic functions

Fundamental group and analytic disks

Let $W$ be a domain in a connected complex manifold $M$ and $w_0\in W$. Let ${\mathcal A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline D$ into $M$ that are holomorphic on the interior of $\overline D$, $f(\partial\mathbb D)\subset W$ and $f(1)=w_0$. On the homotopic equivalence classes $\eta_1(W,M,w_0)$ of ${\mathcal A}_{w_0}(W,M)$ we introduce a binary operation $\star$ so that $\eta_1(W,M,w_0)$ becomes a semigroup and the natural mappings $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ and $\delta_1:\,\eta_1(W,M,w_0)\to\pi_2(M,W,w_0)$ are homomorphisms. \par We show that if $W$ is a complement of an analytic variety in $M$ and if $S=\delta_1(\eta_1(W,M,w_0))$, then $S\cap S^{-1}=\{e\}$ and any element $a\in\pi_2(M,W,w_0)$ can be represented as $a=bc^{-1}=d^{-1}g$, where $b,c,d,g\in S$. \par Let ${\mathcal R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline D$ into $M$ such that $f(\partial{\mathbb D})\subset W$ and $f(1)=w_0$. We describe its open dense subset ${\mathcal R}^{\pm}_{w_0}(W,M)$ such that any connected component of ${\mathcal R}^{\pm}_{w_0}(W,M)$ contains at most one connected component of ${\mathcal A}_{w_0}(W,M)$.

Gleason parts and point derivations for uniform algebras with dense invertible group

It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese $X$ such that $R(X)$ has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in ${\bf C}^N$ is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.