Hyperbolic curvature of holomorphic level curves

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 $f(0)\ne 0$, 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.

To appear in J. Math. Anal. Appl

References in corpus (1)

• Hyperbolic convexity of holomorphic level sets