Closed magnetic geodesics on closed hyperbolic Riemann surfaces
A magnetic geodesic is a curve on a Riemannian manifold moving according to the usual geodesic equations modified by including a magnetic force term. The author proves the existence of Alexandrov embedded (a fortiori closed and null homotopic) magnetic geodesics with any sufficiently small kinetic energy, on any closed surface with sufficiently large negative curvature and sufficiently large positive magnetic force. The method uses an \(S^1\)-equivariant index formula for a certain vector field on an infinite dimensional Sobolev space of loops, and a compactness theorem for the space of Alexandrov embedded magnetic geodesic loops.
scientific article; zbMATH DE number 6073827