Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane

Jin Sun

Published 2023-04-18Version 1

In this paper we study the upper bound of the curvature estimate for nodal sets of harmonic functions in the plane. Using the complex methods, we prove that at any non-critical point $p$, the curvature of any nodal curve of a harmonic function $u$ is upper bounded by $$ \left|\kappa(u)(p)\right|\leq \frac{C}{r} $$ where $u$ has only one nodal curve in $B_r(p)$ across $p$. L. De Carli and Steve M. Hudson proved that the constant $C\leq 24$. In this paper, we prove that the sharp upper bound $C$ is 8, and we also prove that the equality holds if and only if $u$ is a harmonic function related to some Koebe function. On the other hand, we obtain the curvature estimate of nodal curves of harmonic functions at critical points. Thus we prove that, for harmonic functions, the curvature of every nodal curve at any point $p$ is upper bounded by the distance between $p$ and other nodal curves, and the distance from $p$ to the boundary of the domain.