Vol. 38, No. 2, 1971

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ISSN: 0030-8730
The polynomial hull of a thin two-manifold

Michael Benton Freeman

Vol. 38 (1971), No. 2, 377–389

Let M be a smooth real two dimensional submanifold of C2. Near a point of M where its tangent space is complex linear it may, after a certain local biholomorphic change of coordinates, be represented as the graph near 0 in C2 of a smooth function

f(z) = Q(z)+ o(|z|2)

in which z is the first coordinate of C2 and Q is a real valued quadratic form. This paper is concerned with the polynomially convex hull of a small compact set K in M near 0 and associated descriptions of the Banach algebra P(M) of continuous functions uniformly approximable on K by polynomials in two complex variables. It treats the very special case where f has rank 1 near 0 (as an R2-valued map). It is shown that if Q has nonzero eigenvalues of opposite sign, then all sufficiently small compact sets K are polynomially convex, and P(K) is the full algebra of continuous functions. If Q has nonzero eigenvalues of the same sign, the polynomial hull of K is described in terms of a foliation by certain simple analytic sets in C2, and P(K) is isomorphic to the algebra of continuous functions on the hull whose restriction to the interior of each analytic set is holomorphic.

Mathematical Subject Classification 2000
Primary: 32E30
Received: 4 December 1970
Revised: 29 March 1971
Published: 1 August 1971
Michael Benton Freeman