For X a compact set in
C2,h(X) denotes the polynomially convex hull of X. We are concerned with the
existence of analytic varieties in h(X)∖X.X is called “invariant” if (z,w) in X
implies (ei𝜃z,e−i𝜃w) is in X, for all real 𝜃.X is called an “invariant disk” if
there is a continuous complex-valued function a defined on 0 ≦ r ≦ 1 with
o(0) = a(1) = 0, such that X = {(z,w)∣|z|≦ 1,w = a(|z|)∕z}. Let X be an
invariant set and put f(z,w) = zw. Let Ω be an open disk in C ∖ f(X) and put
f−1(Ω) = {(z,w) in h(X)|zw ∈ Ω}. In Theorem 2 we show that if f−1(Ω) is not
empty, then f−1(Ω) contains an analytic variety. Let now X be an invariant disk,
with certain hypotheses on the function 0. Then we show in Theorem 3
that f−1(Ω) is the union of a one-parameter family of analytic varieties.
A key tool in the proofs is a general subharmonicity property of certain
functions associated to a uniform algebra. This property is given in Theorem
1.
