A Zoll counterexample to a geodesic length conjecture
Differential Geometry
2014-10-03 v2 Functional Analysis
Metric Geometry
Abstract
We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D.
Cite
@article{arxiv.0711.1229,
title = {A Zoll counterexample to a geodesic length conjecture},
author = {Florent Balacheff and Christopher Croke and Mikhail G. Katz},
journal= {arXiv preprint arXiv:0711.1229},
year = {2014}
}
Comments
10 pages; to appear in Geometric and Functional Analysis