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We solve the geodesic equation in the space of K\"ahler metrics under thesetting of asymptotically locally Euclidean (ALE) K\"ahler manifolds and weprove global $\mathcal{C}^{1,1}$ regularity of the solution. Then, we relatethe solution of the geodesic equation to the uniqueness of scalar-flat ALEmetrics. To this end, we study the asymptotic behavior of$\varepsilon$-geodesics at spatial infinity. Under the assumption that theRicci curvature of a reference ALE K\"ahler metric is non-positive, convexityof the Mabuchi $K$-energy along $\varepsilon$-geodesics. However, we will alsoprove that on the line bundle $\mathcal{O}(-k)$ over$\mathbb{C}\mathbb{P}^{n-1}$ with $n \geq 2$ and $k \neq n$, no ALE K\"ahlermetric can have non-positive (or non-negative) Ricci curvature.

Geodesic Equations on asymptotically locally Euclidean K\"ahler manifolds