Approximation of Singularity Sets with Analytic Graphs over the Ball in C

Let h be a smooth function on the ball in C2 whose gradient has length less than or equal to 1. We show that if h is uniformly near an analytic function on every complex affine one-dimensional slice then it must be near some function analytic on the whole ball. We use this to show the following: a singularity set over the ball which is near the graph of a function h with |∇h| ≤ 1 must be near the graph of some analytic function over the ball. Let B2 be the open unit ball in C , let S = ∂B2 and let K be a compact subset of B2×C. We say that K is a singularity set if (B2×C)\K is pseudoconvex. This implies that there exists an f analytic on (B2×C)\K which is singular at each point of the boundary of K in B2×C. If we let ∆ denote the closed unit disk in C then we can make a similar definition of a singularity set in (int∆)×C. Singularity sets were studied as early as 1909 by Hartogs [3] and later by Oka [5] and Nishino [4]. One issue that has been studied is the question of whether such sets possess analytic structure, i.e., whether they contain analytic varieties. Wermer [9] and S lodkowski [8] showed that such an expectation is reasonable in general by proving a maximum modulus principle for singularity sets K projecting onto ∆; in particular, the following holds: Proposition. If (z0, w0) ∈ K then for every polynomial Q, |Q(z0, w0)| ≤ sup (z,w)∈K∩{|z|=1} |Q(z, w)|. (See (1) on p. 264 of [1].) In [1], Alexander and Wermer showed that a singularity set projecting onto the disk which is reasonably “thin” must be near an analytic graph. More precisely, they showed Theorem of [1]. Let λ → a(λ) be a continuous function defined for |λ| ≤ 1 with |a(λ)| ≤ 1 for all λ. Fix r > 0. Suppose that there exists a singularity set X projecting onto ∆ such that X is contained in the tube {(λ,w) ∣∣ |w − a(λ)| < r}. Then there exists an analytic function λ → f(λ) such that |f(λ) − a(λ)| ≤ 4r for each λ in the unit disk. Received by the editors May 17, 1996. 1991 Mathematics Subject Classification. Primary 32E30, 32F15.