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Approximation of singularity sets with analytic graphs over the ball in $\mathbf {C}^2$

Author: Marshall A. Whittlesey
Journal: Proc. Amer. Math. Soc. 125 (1997), 3259-3265
MSC (1991): Primary 32E30, 32F15
MathSciNet review: 1415342
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Abstract: Let $h$ be a smooth function on the ball in C$^{2}$ 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 $|\nabla h|\leq 1$ must be near the graph of some analytic function over the ball.

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Additional Information

Marshall A. Whittlesey
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Singularity set, analytic structure
Received by editor(s): May 17, 1996
Additional Notes: This work is part of the author’s Ph.D. thesis and was supported in part by the R. B. Lindsay Graduate Fellowship. The author would also like to express his appreciation for the guidance of his thesis advisor John Wermer
Communicated by: Eric Bedford
Article copyright: © Copyright 1997 American Mathematical Society