Flat points in zero sets of harmonic polynomials and harmonic measure from two sides

TL;DR

This paper provides quantitative estimates on the local flatness of zero sets of harmonic polynomials, offering insights into their geometric structure and implications for free boundary problems involving harmonic measure from two sides.

Contribution

It introduces a dichotomy for zero sets of harmonic polynomials regarding their flatness at different scales and applies this to free boundary problems with harmonic measure.

Findings

Zero sets are either uniformly far from hyperplanes or become arbitrarily flat at small scales.

Quantitative estimates of flatness are established for zero sets of harmonic polynomials.

Application to free boundary problems shows the relevance of zero set geometry in harmonic measure analysis.

Abstract

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far away from a hyperplane in the Hausdorff distance at all scales or the zero set becomes locally flat on small scales with arbitrarily small constant. An application is given to a free boundary problem for harmonic measure from two sides, where blow-ups of the boundary are zero sets of harmonic polynomials.