Vol. 58, No. 1, 1975

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Subharmonicity and hulls

John Wermer

Vol. 58 (1975), No. 1, 283–290

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,ei𝜃w) 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 f1(Ω) = {(z,w) in h(X)|zw Ω}. In Theorem 2 we show that if f1(Ω) is not empty, then f1(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 f1(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.

Mathematical Subject Classification 2000
Primary: 32E20
Secondary: 46J20
Received: 23 April 1974
Published: 1 May 1975
John Wermer