Geodesic Equations on asymptotically locally Euclidean Kähler manifolds

In this paper, we solve the geodesic equation in the space of Kähler metrics under the setting of asymptotically locally Euclidean (ALE) Kähler manifolds and establish global C1,1 regularity of the solution. The solution of the geodesic equation is then related to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of ε-geodesics at spatial infinity. We will prove the convexity of Mabuchi K energy along ε-geodesics under the assumption that the Ricci curvature of a reference ALE Kähler metric is non-positive. However, by testing the Ricci curvature of ALE Kähler metrics, we find that on the line bundle O(-k) over CPn-1 with n≥2 and k≠n, all ALE Kähler metrics cannot have non-positive (or non-negative) Ricci curvature.