Polynomial hull of a torus fibered over the circle

Given a 2-sheeted torus over the circle with winding number 1, we prove that its polynomial hull is a union of 2-sheeted holomorphic discs. Moreover, when the hull is non-degenerate its boundary is a Levi-flat solid torus foliated by such discs.

POLYNOMIAL HULLS AND ENVELOPES OF HOLOMORPHY OF SUBSETS OF STRICTLY PSEUDOCONVEX BOUNDARIES

First we give a construction of a Cantor set in the unit sphere in ℂ3 the polynomial hull of which contains interior points. Such sets were known to exist in spheres in ℂ2. Secondly, we construct an open connected subset of the unit sphere in ℂ3 with infinitely sheeted envelope of holomorphy.

POLYNOMIAL APPROXIMATION, LOCAL POLYNOMIAL CONVEXITY, AND DEGENERATE CR SINGULARITIES — II

We provide some conditions for the graph of a Hölder-continuous function on [Formula: see text], where [Formula: see text] is a closed disk in ℂ, to be polynomially convex. Almost all sufficient conditions known to date — provided the function (say F) is smooth — arise from versions of the Weierstrass Approximation Theorem on [Formula: see text]. These conditions often fail to yield any conclusion if rank ℝDF is not maximal on a sufficiently large subset of [Formula: see text]. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in ℂ2 at an isolated complex tangency.

Analytic discs in the polynomial hull of a disc fibration over the sphere

It is shown that for each point P in the interior of the polynomial hull of a disc fibration X over the unit sphere ∂n there exists an H∞ analytic disc with boundary in X and passing through p.

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

Consider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.