A Riemannian Bieberbach estimate

Details A Riemannian Bieberbach estimate A Riemannian Bieberbach estimate

2009-05-15

arxiv.org/abs/0905.2604v1

Mathematics

Differential Geometry

Scientific paper

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.

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