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Deforming an є-close-to-hyperbolic metric to a hyperbolic metric

Part of: Global differential geometry

Published online by Cambridge University Press:  18 April 2018

Pedro Ontaneda
Pedro Ontaneda*
Affiliation:
Department of Mathematical Sciences, Binghamton University, PO Box 6000, Binghamton, NY 13902-6000, USA (pedro@math.binghamton.edu)
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Abstract

We show how to deform a metric of the form g = gr + dr2 to a metric = Hr + dr2, which is a hyperbolic metric for r less than some fixed λ, and coincides with g for r large. Here by hyperbolic metric we mean a metric of constant sectional curvature equal to -1. We study the extent to which is close to hyperbolic everywhere, if we assume g is close to hyperbolic. A precise definition of the close to hyperbolic concept is given. We also deal with a one-parameter version of this problem. The results in this paper are used in the problem of smoothing Charney–Davis strict hyperbolizations.

Keywords

hyperbolic metricmetric deformationhyperbolization

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 629 - 641
Copyright
Copyright © Royal Society of Edinburgh 2018 

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