A Riemannian Bieberbach estimate

TL;DR

This paper extends the classical Bieberbach estimate from complex analysis to higher-dimensional conformal immersions, incorporating geometric and dynamical correction terms, motivated by minimal surface theory.

Contribution

It introduces a generalized Bieberbach inequality for injective smooth conformal immersions into higher dimensions, involving second fundamental form and Riemannian dynamical quantities.

Findings

Derived a new inequality with geometric and dynamical correction terms.

Connected the estimate to properties of conformal attractors.

Motivated by conjectures in embedded minimal surface theory.

Abstract

The Bieberbach estimate, a pivotal result in the classical theory of univalent functions, states that any injective holomorphic function $f$ on the open unit disc $D$ satisfies $|f"(0)|\leq 4 |f'(0)|$. We generalize the Bieberbach estimate by proving a version of the inequality that applies to all injective smooth conformal immersions $f : D\to \Bbb R^n, n\geq 2$. The new estimate involves two correction terms. The first one is geometric, coming from the second fundamental form of the image surface $f(D)$. The second term is of a dynamical nature, and involves certain Riemannian quantities associated to conformal attractors. Our results are partly motivated by a conjecture in the theory of embedded minimal surfaces.