A Riemannian Bieberbach estimate

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)| ≤ 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 → R n , n ≥ 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.

Conformal geometry, Euler numbers, and global invertibility in higher dimensions

“…There are reasons to believe that there may be a common thread between these two statements. A Riemannian Bieberbach estimate [15] might be a starting point. Indeed, applying the classical Bieberbach estimate |f (0)| ≤ 4|f (0)|, valid for injective holomorphic maps in the unit disc, to suitable scalings of an injective entire function g, one can show that g must be an affine linear map.…”

“…What is lacking in the study of parabolic simply-connected embedded minimal surfaces is a well-chosen application of [15], together with a suitable scaling process that, in the limit, gives rise to only ruled minimal surfaces. By a classical theorem of Catalan, these are planes and helicoids.…”

Complete densely embedded complex lines in ℂ²

“…For example, if a > 0 is irrational then the map C ∋ z → (e z , e az ) ∈ C 2 is an injective immersion, but the image of the negative real axis is a curve of finite length in C 2 terminating at the origin. On the other hand, it is an open problem whether a conformal minimal embedding C → R 3 is necessarily proper; see [11,Conjecture 1.2].…”

Finiteness of prescribed fibers of local biholomorphisms: a geometric approach

“…The present paper is part of a larger program whose aim is to study global invertibility and related questions in the complex-analytic setting, using geometric, analytic and topological tools ( [3], [8], [9], [14]). …”