Wermer examples and currents

TL;DR

This paper constructs the first examples of positive closed currents in complex two-space with continuous potentials that are not laminar, supported on sets lacking analytic structure, especially near $C^2$ regularity.

Contribution

It provides the first known examples of such currents with continuous potentials and no laminar structure, challenging previous expectations about laminarity at high regularity.

Findings

Constructed positive closed currents with continuous potentials

Currents supported on sets without analytic structure

Examples are $C^{1,eta}$ for all $eta<1$

Abstract

In this paper we give the first examples of positive closed currents in $\mathbb{C}^2$ with continuous potentials, vanishing self-intersection, and which are not laminar. More precisely, they are supported on sets "without analytic structure". The result is mostly interesting when the potential has regularity close to $C^2$, because laminarity is expected to hold in that case. We actually construct examples which are $C^{1,\alpha}$ for all $\alpha<1$.