In this talk, we will establish a complete picture of metric geometry of Calabi-Yau metrics in complex dimension two. The main part will focus on the resolution of the following three well-known conjectures in the field.

(1) Any volume collapsed limit of unit-diameter K3 metrics is isometrically classified as: the quotient of a flat 3D torus by an involution, a singular special Kähler metric on the topological 2-sphere, or the unit interval.

(2) Any complete non-compact hyperkähler 4-manifold with quadratically integrable curvature, must have a classified model end.

(3) Any gravitational instanton can be biholomorphically compactified to be an open dense subset of certain compact algebraic surface.