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 $\kappa \le 8$ 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.

arXiv commentThis paper is withdrawn: the result has been proven earlier by Kuran (On the zeros of harmonic functions, J. London Math. Soc. 44, 1969, 303-309)