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On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

Published online by Cambridge University Press:  20 November 2018

Simon Brendle  and
Otis Chodosh
Simon Brendle
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA 94305 e-mail: sbrendle@stanford.eduochodosh@stanford.edu
Otis Chodosh
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA 94305 e-mail: sbrendle@stanford.eduochodosh@stanford.edu
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Abstract

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Motivated by Almgren’s work on the isoperimetric inequality, we prove a sharp inequality relating the length and maximum curvature of a closed curve in a complete, simply connected manifold of sectional curvature at most −1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function defined on pairs of points.

Keywords

53C20manifoldcurvature
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 713 - 722
Copyright
Copyright © Canadian Mathematical Society 2015

References

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