Riemann surfaces in fibered polynomial hulls.
Abstract
Let D be the closed unit disk in C , let T be the circle, let
P:D × C -->D be projection, and let A(D) be the algebra of
complex functions continuous on D and analytic in int D. Let E be a
compact set in C2 such that P(E)=T, and let Ez
={w | (z,w) is in E}. Suppose further that
(a) for every z in T, Ez is the
union of 2 nonempty disjoint connected compact sets with connected
complement;
(b) there exists a
function Q(z,w)=(w-R(z))2-S(z) quadratic in w with R,S in A(D) such
that for all z in T, {w | Q(z,w)=0} is contained in int Ez where S
has only one zero in int D, counting multiplicity;
(c) for every z
in T, the map w |--->Q(z,w) is injective on each component of Ez.
Then we prove that Ê \ E is the union of analytic
disks 2-sheeted over int D, where Ê is the polynomial
convex hull of E. Furthermore, we show that the part of boundary of
Ê which sits over int D is the disjoint union of such disks.