This monograph investigates the geometric and analytic properties of geodesic rays within the infinite-dimensional space of Kähler potentials H associated with a compact Kähler manifold (X, ). We rigorously analyze the correspondence between these rays and solutions to the homogeneous complex Monge-Ampère equation (HCMA) on the product space X , treating the equation as a boundary value problem with degenerate boundary data. Special attention is given to the regularity of weak solutions, specifically the C1,1 estimates derived from the maximum principle for the Laplacian of the potential. Furthermore, we explore the asymptotic behavior of the Mabuchi K-energy functional along these rays, establishing a link between the solvability of the degenerate HCMA and the algebraic notion of K-stability. The analysis extends to the construction of geodesic rays associated with test configurations, providing a differential-geometric framework for the Yau-Tian-Donaldson conjecture. We prove that under suitable convexity conditions, the weak geodesic rays exhibit sufficient regularity to define the asymptotic slope of the K-energy, thereby characterizing the obstruction to the existence of constant scalar curvature Kähler (cscK) metrics.