Volumes of trajectory-balls for Kähler magnetic fields (Q742529)

A closed 2-form on a Riemannian manifold can be seen as a magnetic field. One can then add terms into the geodesic equation of the manifold to incorporate the influence of the magnetic field, to obtain ``magnetic trajectories''. On a Kähler manifold, the Kähler form provides a natural example of a magnetic field. The authors alter the geodesic equation of a Kähler manifold by adding terms to incorporate its Kähler form. They then study the deformed exponential map, using magnetic trajectories in place of usual geodesics. The image of a Euclidean ball in a tangent space under this deformed exponential map they call a \textit{trajectory ball}. The authors estimate volumes of trajectory balls in any Kähler manifold, from above and below, in terms of bounds on sectional and Ricci curvature.