On the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces

Abstract

We determine the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces for (i) arbitrary constant magnetic fields on the two-sphere and (ii) strong constant magnetic fields for higher genus surfaces. On $S^2$ we further give an explicit $\text{SO}(3)$-equivariant compactification of the twisted tangent bundle to $S^2\times S^2$ with split symplectic form. The former is the phase space of a charged particle moving on the two-sphere in a constant magnetic field, the latter is the configuration space of two massless coupled angular momenta.

The results of this paper are now contained in arXiv:2311.00467 for the part about twisted tangent bundles over surfaces and in paper arXiv:2306.11382 for the part on the symplectomorphism between the twisted tangent bundle of $S^2$ and $S^2\times S^2$ with split symplectic form. The proof presented in arXiv:2311.00467 is very different and much shorter than the proof in this draft

References in corpus (1)

• On the existence of periodic orbits for magnetic systems on the two-sphere