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# On the Maximum Curvature of Closed Curves in Negatively Curved Manifolds

Published:2015-04-30
Printed: Dec 2015
• Simon Brendle,
Department of Mathematics , Stanford University , 450 Serra Mall, Bldg 380 , Stanford, CA 94305
• Otis Chodosh,
Department of Mathematics , Stanford University , 450 Serra Mall, Bldg 380 , Stanford, CA 94305
 Format: LaTeX MathJax PDF

## Abstract

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: manifold, curvature
 MSC Classifications: 53C20 - Global Riemannian geometry, including pinching [See also 31C12, 58B20]

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