Analytic Discs Research Articles

Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$

Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$

Polynomially convex hulls with piecewise smooth boundaries

is the boundary of a compact convex set Y(z) c C m with nonempty interior. Set Y= U {z} x Y(z). Alexander and Wermer [1] and Slodkowski [8] proved zcbD independently that each point p in the polynomial convex hull ~ of M, n (p) ~D, lies in the graph of a bounded analytic disk F(z) = ( z , f 1 (z) , . . . ,f~ (z)) ( f j~ H | (D) for l < j < m) with image in 3~r. Alexander and Wermer obtained a more precise information in the case when m = 1 and each fiber M(z) is a circle [1, Theorems 2 and 3]. In this paper we shall assume the existence of an analytic disk F o (z) = (z, f0 (z)) on D, continuous on / ) , such thatfo (z) is an interior point of Y(z) for each z ~ bD. Then each bounded analytic disk F ( z ) = (z , f (z)) in AI which passes through a boundary point of A~r lying over D is entirely contained in the boundary o fAir and its cluster set F(D) \ F ( D ) is contained in M (Theorem 1.1). If m = 1 and M is a smooth submanifold of r as above, then the hull 3~r has piecewise smooth boundary and is f'dled by the images of analytic disks with smooth boundary values contained in M (Theorem 1.2). We also obtain a uniqueness result for the representing measures (Corollary 1.3).

On the Hans Lewy extension phenomenon in higher codimension

In this work the authors extend the results of their paper entitled Families of analytic discs in C n {{\mathbf {C}}^n} with boundaries on a prescribed C R CR submanifold. It is proven that a nongeneric CR manifold M M whose Lewy form is not identically zero can be extended to a manifold M ~ \tilde M of one higher dimension, which is foliated by analytic discs. Moreover this result is used to prove that a sufficiently smooth CR function f f on M M extends to a function f ~ \tilde f which is CR on M ~ \tilde M .

Spectra of Invariant Uniform and Transform Algebras

For $G$ a locally compact abelian group, any closed invariant proper subalgebra of ${C_0}(G)$ has analytic discs in its spectrum. Related results are given for $A(G)$ and $B(G)$.

Spectra of invariant uniform and transform algebras

For G G a locally compact abelian group, any closed invariant proper subalgebra of C 0 ( G ) {C_0}(G) has analytic discs in its spectrum. Related results are given for A ( G ) A(G) and B ( G ) B(G) .

Parts in $H\sp{\infty }$ with homeomorphic analytic maps

Hoffman characterized the parts of ${H^\infty }$ as either singleton points or analytic discs. He showed that a part belongs to the latter category if and only if it is hit by the closure of an interpolating sequence and that there are cases where a corresponding analytic map is a homeomorphism and cases where it is not. We show that there is no class, $\mathcal {C}$, of subsets of the open unit disc such that an analytic map of a part $P$ is a homeomorphism if and only if $P$ is hit by the closure of some set in $\mathcal {C}$.

Famiglie di dischi analitici con bordo su sottovarietàCR non generiche

In this work the authors prove that all sufficiently small analytic discs in the tangent space of a non generic embedded CR manifold M′⊂CN (M′ at least of class C4) can be lifted uniquely to analytic discs in CN with boundaries on M′ Moreover if M′ is real analytic or C∞ then real analytic or C∞ discs lift without loss of derivatives. If M′ is of class CK then there is a 1+ε loss of derivatives in the lifting. A stability result is also proven.

Point derivations in certain sup-norm algebras

1. Let A be a closed point-separating subalgebra of C(X) containing the constants, where X is a compact Hausdorff space. MA will denote the space of multiplicative linear functionals p on A, and to each such p we associate its kernel A,. The A, are precisely the maximal ideals of A. Under certain hypotheses, it is known that analytic discs can be embedded in MA. Wermer [W1] showed that if A is a Dirichlet algebra on X, then each Gleason part of A is either a single point or an analytic disc. Hoffman [H] then generalized Wermer's result to logmodular algebras. Finally Lumer [L] observed that the conclusion is really local: if p has a unique representing measure on X, then the part for A containing p consists either of p alone or of an analytic disc. Our objective in this paper is to take the weakest of these possible hypotheses, namely Lumer's, and show that in a broader sense the analytic disc at p, if there is one, really does account for all the analytic structure at (p. Specifically, we show that all the bounded derivations and higher derivatives of A at cp are just differentiations with respect to the analytic structure of the analytic disc.

Analytic embedding and mean square approximation

1. Introduction. Our results concern the generalized Hp spaces derived from certain function algebras, particularly those spaces associated with analytic discs embedded in the carrier space. Such general Hp spaces are obtained by considering a uniform algebra =A(X), fixing a measure monX representing a multiplicative linear functional on A, and defining H(dm) as the closure of in Lv(dm) (w* closure for p= ). One further condition (the unique representing measure property) is imposed on so that a large portion of the classical theory concerning the Hardy classes remains valid for H*(dm). Our first result provides an explicit determination of the analytic embedding mapping of Wermer [7] (of the unit disc onto the Gleason part associated with ra) in terms of the outer function associated with a certain Radon-Nikodym derivative. We then use this explicit expression to generalize a result of Szegb' on mean square approximation of the function 1 by maximal ideals of functions, relating two different homomorphisms in the same Gleason part. 2. Preliminaries. If A is a compact Hausdorff space, a uniformly closed algebra A=A(X) of continuous complex-valued functions is

A Cantor set whose polynomial hull contains no analytic discs

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in $\mathbb{C}^3$ whose polynomial hull is strictly larger than $X$ but contains no analytic discs.

On a family of analytic discs attached to a real submanifold $M ⊂ \mathbb{C}^{N+1}$

On a family of analytic discs attached to a real submanifold $M ⊂ \mathbb{C}^{N+1}$