On the curvature of level sets of harmonic functions

TL;DR

This paper establishes a sharp upper bound of 8 on the curvature of level sets of harmonic functions in the unit disk, characterizing the extremizer and its uniqueness.

Contribution

It proves the sharp curvature bound for harmonic level sets in the disk and identifies the unique extremizer, advancing understanding of harmonic function geometry.

Findings

Curvature bound for harmonic level sets is exactly 8.

The extremizer achieving the bound is unique up to symmetries.

The result is sharp and optimal for the class of harmonic functions considered.

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.