A riemannian bieberbach estimate

Francisco Fontenele Universidade Federal Fluminense Frederico Xavier University of Notre Dame

Differential Geometry mathscidoc:1609.10176

j. differential geometry, 85, (1), 1-14, 2010
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 → Rn, 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.
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  author={Francisco Fontenele, and Frederico Xavier},
  booktitle={j. differential geometry},
Francisco Fontenele, and Frederico Xavier. A RIEMANNIAN BIEBERBACH ESTIMATE. 2010. Vol. 85. In j. differential geometry. pp.1-14. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911214437115570833.
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