Distances From Unbounded Trajectories to Their Limit-Strings on a Hadamard Kähler Manifold

When sectional curvatures of a Hadamard Kahler manifold are not greater than c, every trajectory half-line $$\gamma $$ for a Kahler magnetic field of strength not greater than $$\sqrt{|c|}$$ is unbounded and have limit point $$\gamma (\infty )$$ in the ideal boundary. We take a geodesic half-line $$\sigma $$ whose origin is the origin of $$\gamma $$ and the limit point coincides with $$\gamma (\infty )$$ . We give an estimate of the distance from $$\gamma (t)$$ to $$\sigma $$ and show that its growth is not greater than linear order.