Riemannian two-sphere contains two smoothly immersed curves ...

Problem's Description

The text below is excerpted from [1], with minor modifications in wording.

Birkhoff's celebrated theorem states that any two-sphere equipped with a Riemannian metric invariably harbors a closed geodesic. An enduring puzzle posed by V.I. Arnold in 1981 [2] is as follows:

Conjecture 1.1 (Arnold) Every Riemannian two-sphere contains two smoothly immersed curves with curvature $c$ for any $c$.

Curves with curvature $c$ represent closed orbits of charged particles navigating magnetic fields. Investigating these magnetic orbits has spanned various fields, including Morse-Novikov theory, variational functionals, dynamical systems, and Aubry-Mather theory.

Even two-spheres with arbitrary metrics boast a minimum of three embedded closed geodesics. For convex two-spheres, Calabi-Cao's min-max argument guarantees at least one.

A related conjecture by S. Novikov concerning constant curvature asserts:

Conjecture 1.2 (Novikov) Every Riemannian two-sphere contains a smoothly embedded curve with curvature $c$ for any $c$.

Immersed curves with constant curvature $c$ have been confirmed to exist on two-spheres with both large and small curvatures, as well as on nearly spherical ones. For exceedingly large curvatures, infinitesimal geodesic circles of high curvature can always be found. When curvature approaches zero, the implicit function theorem facilitates the discovery of constant curvature curves near non-degenerate closed geodesics.

Schneider's investigation on positively curved spheres reveals the existence of at least two immersed constant curvature curves for each curvature value. The challenge in this context lies in achieving embeddedness. Schneider's remarkable result demonstrates that 1/4-pinched surfaces invariably contain two embedded curves of any curvature $c$.

The authors in [1] present evidence of closed curves on any surface, not just two-spheres, where the curvature alternates between $c$ and $c'$. Our curves also possess near-embeddedness properties.

References:

[1] Arnold, V.I.: Arnold’s Problems. Springer, Berlin, Translated and revised edition of the 2000 Russian original, with a preface by V. Philippov, A. Yakivchik and M. Peters (2004)

[2] Calabi, E., Cao, J.: Simple closed geodesics on convex surfaces. J. Diff. Geom. 36(3), 517–549 (1992)

[3] Novikov, S.P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Uspekhi Mat. Nauk 37(5), 1 (1982)

[4] Schneider, M.: Alexandrov embedded closed magnetic geodesics on $S^2$. Ergod. Th. and Dynam. Sys. 32, 1471–1480 (2012)

[5] Schneider, M.: Closed magnetic geodesics on $S^2$. J. Diff. Geom. 87, 343–388 (2011)

1. ArticleIs an originOn the existence of closed $C^{1,1}$ curves of constant curvature

   • Daniel Ketover,
   • Yevgeny Liokumovich

   Calculus of Variations and Partial Differential Equations 62, Paper No. 246, 13, 2023fulltext

2. Book Arnold’s problems

   • V. I. Arnold

   Springer 2004