CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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

© Canadian Mathematical Society, 2017 : https://cms.math.ca/