Polynomial hulls of arcs and curves

Abstract

It is shown that there exist arcs and simple closed curves in C 3 with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded Runge domain of holomorphy in C N (N ≥ 2) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through Cantor sets.2010 Mathematics Subject Classification. Primary 32E20; Secondary 32A38, 32E30, 46J10, 46J15.

Asymptotic first boundary value problem for holomorphic functions of several complex variables

“…We would like to thank the authors of [6]; our Lemma 13 is inspired by a similar result from their manuscript. We would also like to thank Alexander Izzo for drawing our attention to his lovely paper [7], which, as we remarked, makes it possible to simplify our proof of Theorem 11 in the case of Lebesgue measure. We would also like to thank Anush Tserunyan for stimulating discussions regarding descriptive set theory.…”

“…Therefore, E is the desired closed set and f is the desired entire function. ∎ Remark For Lebesgue measure, the proof of Theorem 12 can be somewhat simplified, taking into account Lemma 2.4 in [7], which states that for an arbitrary polynomially convex compact set Y ⊂ C n and ε > 0, there is a totally disconnected polynomially convex compact set K ⊂ Y, such that m(Y/K) < ε and also taking into account, that polynomials are dense in C(K), when K is a totally disconnected polynomially convex compact set [1, Chapter 8, p. 48].…”

Asymptotic first boundary value problem for holomorphic functions of several complex variables

“…Remark. For Lebesgue measure, the proof of Theorem 11 can be somewhat simplified, taking into account Lemma 2.4 in [8] which states that for an arbitary polynomially convex compact set Y ⊂ C n and ǫ > 0, there is a totally disconnected polynomially convex compact set K ⊂ Y, such that m(Y \ K) < ǫ and also taking into account, that polynomials are dense in C(K), when K is a totally disconnected polynomially convex compact set [1, Chapter 8, page 48].…”

“…We thank the authors of [6], our Lemma 13 is inspired by a similar result from their manuscript. We also thank Alexander Izzo for drawing our attention to his lovely paper [8], which, as we remarked, makes it possible to simplify our proof of Theorem 11 in the case of Lebesgue measure. We thank Anush Tserunyan for stimulating discussions regarding descriptive set theory.…”