Harmonic functions with highly intersecting zero sets

TL;DR

This paper demonstrates that harmonic maps from  to  can have an arbitrarily fast growth in the number of isolated zeros within a radius, while maintaining controlled maximal modulus, challenging previous expectations.

Contribution

It introduces a novel construction showing the potential for rapid zero set growth in harmonic maps, extending the understanding of their zero set behavior.

Findings

Number of zeros can grow arbitrarily fast with radius

Maximal modulus growth remains controlled

Analogous to Cornalba-Shiffman counterexamples

Abstract

We show that the number of isolated zeros of a harmonic map $h:\mathbb{R}^2\to \mathbb{R}^2$ inside the ball of radius $r$ can grow arbitrarily fast with $r$, while its maximal modulus grows in a controlled manner. This result is an analogue, in the context of harmonic maps, of the celebrated Cornalba-Shiffman counterexamples to the transcendental B\'{e}zout problem.