Hyperbolic curvature of holomorphic level curves

TL;DR

This paper investigates the hyperbolic curvature of specific level curves of holomorphic functions on the unit disc, establishing bounds, rigidity conditions, and geometric properties related to convexity and curvature.

Contribution

It provides sharp bounds for hyperbolic curvature of level curves, a rigidity theorem characterizing when the curvature vanishes, and bounds on geometric quantities of sublevel sets.

Findings

Hyperbolic curvature bounds for level curves of holomorphic functions.

A rigidity theorem linking zero curvature to automorphisms.

Bounds on hyperbolic area, perimeter, and curvature of sublevel sets.

Abstract

We give sharp bounds for the hyperbolic curvature of the level curve $|z|=|f(z)|$, when $f:\mathbb{D}\to\mathbb{D}$ is holomorphic on the unit disc $\mathbb{D}$ and $f(0)\neq0$, as well as for other related level curves. As a consequence, we point out a rigidity theorem: if the hyperbolic curvature of the above level curve vanishes at some point, then the level curve is a hyperbolic geodesic and $f$ is an automorphism. As another consequence, we prove that $\frac{1}{\sqrt 2}$ is the greatest lower bound of the supremum of $r\in(0,1)$ such that the level curve $|z|=r|f(z)|$ is (Euclidean) convex. This constant turns out to be also the radius of convexity for hyperbolically convex self-maps of $\mathbb{D}$ that fix the origin. We also give (sharp) estimates for the total hyperbolic curvature, hyperbolic area and hyperbolic perimeter of the sublevel sets.