2025/10/31 by Iancu, Mihai, Nechita, Veronica-Oana
#30C35 #30C45 #30F45 #52A10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2511.00227
We give sharp bounds for the hyperbolic curvature of the level curve |z|=|f(z)|, when f:\mathbbD→\mathbbD is holomorphic on the unit disc \mathbbD and f(0)≠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 (1)/(√ 2) is the greatest lower bound of the supremum r∈(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 \mathbbD that fix the origin. We also give (sharp) estimates for the total hyperbolic curvature, hyperbolic area and hyperbolic perimeter of the sublevel sets.