A zoll counterexample to a geodesic length conjecture

Florent Balacheff; Christopher Croke; Mikhail G. Katz

doi:10.1007/s00039-009-0708-9

A zoll counterexample to a geodesic length conjecture

Florent Balacheff, Christopher Croke, Mikhail G. Katz

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

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.

Original language	English
Pages (from-to)	1-10
Number of pages	10
Journal	Geometric and Functional Analysis
Volume	19
Issue number	1
DOIs	https://doi.org/10.1007/s00039-009-0708-9
State	Published - May 2009

Bibliographical note

Funding Information:
Keywords and phrases: Closed geodesic, diameter, Guillemin deformation, sphere, systole, Zoll surface AMS Mathematics Subject Classification: 53C23, 53C22 F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K. supported by the Israel Science Foundation (grants 84/03 and 1294/06)

Funding

Keywords and phrases: Closed geodesic, diameter, Guillemin deformation, sphere, systole, Zoll surface AMS Mathematics Subject Classification: 53C23, 53C22 F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K. supported by the Israel Science Foundation (grants 84/03 and 1294/06)

Funders	Funder number
National Science Foundation	DMS 07-04145, DMS 02-02536
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Israel Science Foundation	1294/06, 84/03

Keywords

Closed geodesic
Diameter
Guillemin deformation
Sphere
Systole
Zoll surface

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keywords = "Closed geodesic, Diameter, Guillemin deformation, Sphere, Systole, Zoll surface",

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AB - 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.

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A zoll counterexample to a geodesic length conjecture

Abstract

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Keywords

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