Wermer type sets and extension of CR functions

TL;DR

This paper constructs special unbounded sets and domains in complex space, demonstrating the extension properties of CR functions and introducing Wermer type sets with unique pseudoconcavity and extension characteristics.

Contribution

It introduces Wermer type sets in higher dimensions and shows how CR functions extend uniquely outside these sets, advancing understanding of complex extension phenomena.

Findings

Existence of unbounded Wermer type sets with no positive-dimensional analytic varieties.

Construction of unbounded strictly pseudoconvex domains with specific CR extension properties.

Demonstration of a CR function extending holomorphically outside a singular set.

Abstract

For each $n\geq2$ we construct an unbounded closed pseudoconcave complete pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety of positive dimension (we call it a \textit{Wermer type set}). We also construct an unbounded strictly pseudoconvex domain $\Omega$ in $\mathbb C^n$ and a smooth $CR$ function $f$ on $\partial\Omega$ which has a single-valued holomorphic extension exactly to the set $\bar\Omega\setminus\mathcal E$.}