A comparison theorem on crescents for Kähler magnetic fields (Q2566556)

A Kähler magnetic field \(\omega\) is a constant multiple of the Kähler 2-form \(\omega_0\) on a Kähler manifold \(M\). If \(\omega=\kappa\omega_0\), a trajectory for \(\omega\) is a curve \(\sigma\) parametrized by arc-length such that \(\nabla_{\dot\sigma}\dot\sigma=\kappa J\dot\sigma\), where \(J\) is the almost complex structure of \(M\). A bow-string for a segment of the trajectory \(\sigma\) is a minimal geodesic joining the extremes of \(\sigma\) and satisfying some additional conditions. In this paper the author proves a comparison theorem for the length of bow-strings for manifolds with sectional curvature bounded from above.