# Minimal discs in hyperbolic space bounded by a quasicircle at infinity

@article{Seppi2014MinimalDI,
title={Minimal discs in hyperbolic space bounded by a quasicircle at infinity},
author={Andrea Seppi},
journal={arXiv: Differential Geometry},
year={2014}
}
• Andrea Seppi
• Published 13 November 2014
• Mathematics
• arXiv: Differential Geometry
We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmuller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichmuller distance between the ends of…
22 Citations

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## References

SHOWING 1-10 OF 32 REFERENCES
On Almost-Fuchsian Manifolds
• Mathematics
• 2009
Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal
Complete noncompact CMC surfaces in hyperbolic 3-space
In this thesis we study the asymptotic Plateau problem for surfaces with constant mean curvature (CMC) in hyperbolic 3-space H3. We give a new, geometrically transparent proof of the existence of a
Counting minimal surfaces in quasi-Fuchsian three-manifolds
• Mathematics
• 2012
It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we
Domains of discontinuity for almost-Fuchsian groups
An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show
Quasi-Fuchsian 3-Manifolds and Metrics on Teichm\
• Mathematics
• 2009
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a
On the Renormalized Volume of Hyperbolic 3-Manifolds
• Mathematics
• 2008
The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present
Minimal surfaces and particles in 3-manifolds
• Mathematics
• 2007
We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the
An extension of Schwarz’s lemma
and has the constant curvature -4. 2. Consider now an analytic function o =f(z) from the circle I zx < 1 to a Riemann surface W. The analyticity is expressed by the fact that every local parameter w
ON UNIFORMIZATION OF RIEMANN SURFACES AND THE WEIL-PETERSSON METRIC ON TEICHMÜLLER AND SCHOTTKY SPACES
• Mathematics
• 1988
A potential is constructed for the Weil-Petersson metric on the Teichm?ller space of marked Riemann surfaces of genus 1\$ SRC=http://ej.iop.org/images/0025-5734/60/2/A03/tex_sm_3170_img2.gif/> in
Quasiconformal Maps and Teichmüller Theory
• Mathematics
• 2006
Preface 1. The Grotzch argument 2. Geometric definition of quasiconformal maps 3. Analytic properties of quasiconformal maps 4. Quasi-isometries and quasisymmetric maps 5. The Beltrami differential