On the curvature of level sets of harmonic functions

Abstract

If a real harmonic function inside the open unit disk $B(0,1) \subset \mathbb{R}^2$ has its level set $\left\{x: u(x) = u(0)\right\}$ diffeomorphic to an interval, then we prove the sharp bound $\kappa \leq 8$ on the curvature of the level set $\left\{x: u(x) = u(0)\right\}$ in the origin. The bound is sharp and we give the unique (up to symmetries) extremizer.