Analytic Discs Research Articles

A hull with no nontrivial Gleason parts

The existence of a nontrivial polynomially convex hull with every point a one-point Gleason part and with no nonzero bounded point derivations is established. This strengthens the \hbox{celebrated} result of Stolzenberg that there exists a nontrivial polynomially convex hull that contains no analytic discs. A doubly generated counterexample to the peak point conjecture is also presented.

Polynomial hulls and analytic discs

The goal of the present note is to construct a class of examples for connected compact sets K subset of C-n whose polynomial hull (K) over cap cannot be covered by analytic discs with boundaries co ...

On the isometries from the unit disk to infinite dimensional Teichmüller spaces

We generalize Earle-Li's polydisk theorem and embedding theorem, and study isometries from the unit disk to infinite dimensional Teichmuller spaces. We also give a simple proof that for any non-Strebel point τ, there exist infinitely many real analytic geodesic disks through τ and the basepoint in infinitely dimensional Teichmuller spaces.

Topology of the maximal ideal space of H∞ revisited

Let M(H∞) be the maximal ideal space of the Banach algebra H∞ of bounded holomorphic functions on the unit disk D⊂C. We prove that M(H∞) is homeomorphic to the Freudenthal compactification γ(Ma) of the set Ma of all non-trivial (analytic disks) Gleason parts of M(H∞). Also, we give alternative proofs of important results of Suárez asserting that the set Ms of trivial (one-pointed) Gleason parts of M(H∞) is totally disconnected and that the Čech cohomology group H2(M(H∞),Z)=0.

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.

Analytic discs, global extremal functions and projective hulls in projective space

Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions, we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in $\mathbb{P}^{n}$. As a corollary, we get a characterization of the projective hull for connected compact sets in $\mathbb{P}^{n}$ by the existence of analytic discs.

Stationary holomorphic discs and finite jet determination problems

We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in \(\mathbb{C }^{n+1}\), which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class \(\mathcal{C }^4\). This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.

Plurisubharmonic subextensions as envelopes of disc functionals

We prove a disc formula for the largest plurisubharmonic subextension of an upper semicontinuous function on a domain W in a Stein manifold to a larger domain X under suitable conditions on W and X. We introduce a related equivalence relation on the space of analytic discs in X with boundary in W . The quotient, if it is Hausdorff, is a complex manifold with a local biholomorphism to X. We use our disc formula to generalise Kiselman’s minimum principle. We show that his infimum function is an example of a plurisubharmonic subextension.

Green versus Lempert functions: A minimal example

The Lempert function for a set of poles in a domain of Cn at a point z is obtained by taking a certain infimum over all analytic disks going through the poles and the point z, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.

Characterizations of projective hulls by analytic discs

The notion of the projective hull of a compact set in a complex projective space $\mathbb{P}^{n}$ was introduced by Harvey and Lawson in 2006. In this paper, we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.

Polynomial hulls and proper analytic disks

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set K defined on the boundary of the unit ball.

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.

Siciak–Zahariuta extremal functions, analytic discs and polynomial hulls

We prove two disc formulas for the Siciak–Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω ⊂ C n . We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n − 2 , then compactness of the Hankel operator H β implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C 2 we show that the condition on β is necessary and sufficient for compactness of H β .

A Morse-theoretical proof of the Hartogs extension theorem

One hundered years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: Holomorphic functions in a connected neighborhood V(∂Ω) of a connected boundary ∂Ω ⋐ℂn ≥ 2) do extend holomorphically and uniquely to the domain ό. Martinelli, in the early 1940’s, and Ehrenpreis in 1961 obtained a rigorous proof, using a new multidimensional integral kernel or a short\(\bar \partial \) argument, but it remained unclear how to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906), and E. E. Levi (1911) in some special, model cases. In fact, known attempts (e.g., Osgood, 1929, Brown, 1936) struggled for monodromy against multivaluations, but failed to get the general global theorem.

Convergence and multiplicities for the Lempert function

Given a domain Ω⊂ℂn, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.

Siciak–Zahariuta extremal functions and polynomial hulls

We use our disc formula for the Siciak–Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs. The polynomial hull K of a compact subset K of complex affine space C n is the compact set of those x ∈ C for which |P (x)| ≤ supK |P | for all complex polynomials P in n variables. The set K is said to be polynomially convex if K = K. Polynomial hulls are usually difficult to determine. By the maximum principle for holomorphic functions it is clear that x ∈ K if there is a continuous map f from the closed unit disc D into C, holomorphic on D, such that f(0) = x and f maps the unit circle into K. In the early days of polynomial convexity theory it seemed possible that K might simply be the union of all analytic discs in C with boundary in K. Stolzenberg’s counterexample of 1963 [7] showed that K \K may be nonempty—and in fact quite large (see [1])—without containing any nonconstant analytic discs. The question of whether polynomial hulls could nevertheless be somehow described in terms of analytic discs remained open for three decades, until Poletsky derived an answer from his theory of disc functionals [6] (see Theorem 2 below). In this note, we give a different characterization of the polynomial hull of a connected compact subset of C, based on our generalization in [5] of Lempert’s disc formula for the Siciak–Zahariuta extremal function from the convex case to the connected case. The gist of both results, Poletsky’s and ours, is to suitably weaken the requirement that the analytic discs in the 2000 Mathematics Subject Classification: Primary 32E20; Secondary 32U35.

Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols

We derive conditions for compactness of Hankel operators H f : A 2 ( Ω ) → L 2 ( Ω ) ( H f ( g ) : = ( I − P ) ( f ¯ g ) ) with bounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω ⊆ D k .

Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs

We prove that homologically nontrivial generic smooth ( 2 n − 1 ) -parameter families of analytic discs in C n , n ⩽ 2 , attached by their boundaries to a CR-manifold Ω, test CR-functions in the following sense: if a smooth function on Ω analytically extends into any analytic discs from the family, then the function satisfies tangential CR-equations on Ω. In particular, we give an answer (Theorem 1) to the following long standing open question, so called strip-problem, earlier solved only for special families (mainly for circles): given a smooth one-parameter family of Jordan curves in the plane and a function f admitting holomorphic extension inside each curve, must f be holomorphic on the union of the curves? We prove, for real-analytic functions and arbitrary generic real-analytic families of curves, that the answer is “yes,” if no point is surrounded by all curves from the family. The latter condition is essential. We generalize this result to characterization of complex curves in C 2 as real 2-manifolds admitting nontrivial families of attached analytic discs (Theorem 4). The main result implies fairly general Morera type characterization of CR-functions on hypersurfaces in C 2 in terms of holomorphic extensions into three-parameter families of attached analytic discs (Theorem 2). One of the applications is confirming, in real-analytic category, the Globevnik–Stout conjecture (Theorem 3) on boundary values of holomorphic functions. It is proved that a smooth function on the boundary of a smooth strictly convex domain in C n extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain. The proofs are based on a universal approach, namely, on the reduction to a problem of propagation, from the boundary to the interior, of degeneracy of CR-foliations of solid torus type manifolds (Theorem 2.2).

CR extension from hypersurfaces of higher type

We prove extension of CR functions from a hypersurface M of C N in presence of the so-called sector property. If M has finite type in the Bloom–Graham sense, then our result is already contained in [C. Rea, Prolongement holomorphe des fonctions CR, conditions suffisantes, C. R. Acad. Sci. Paris 297 (1983) 163–166] by Rea. We think however, that the argument of our proof carries an expressive geometric meaning and deserves interest on its own right. Also, our method applies in some case to hypersurfaces of infinite type; note that for these, the classical methods fail. CR extension is treated by many authors mainly in two frames: extension in directions of iterated of commutators of CR vector fields (cf., for instance, [A. Boggess, J. Pitts, CR extension near a point of higher type, Duke Math. J. 52 (1) (1985) 67–102; A. Boggess, J.C. Polking, Holomorphic extension of CR functions, Duke Math. J. 49 (1982) 757–784. [4]; M.S. Baouendi, L. Rothschild, Normal forms for generic manifolds and holomorphic extension of CR functions, J. Differential Geom. 25 (1987) 431–467. [1]]); extension through minimality towards unprecised directions [A.E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181 (7) (1990) 951–964. [6]; A.E. Tumanov, Analytic discs and the extendibility of CR functions, in: Integral Geometry, Radon Transforms and Complex Analysis, Venice, 1996, in: Lecture Notes in Math., vol. 1684, Springer, Berlin, 1998, pp. 123–141].