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

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.

27 pages, 6 figures (in version 2: updated captions and figures, fixed typos and corrected constant in Lemma 4.1)