We investigate the collapsing geometry of hyperkaehler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperkaehler metrics on the K3 manifold is isometric to one of the follow
ing: the quotient of a flat 3-torus by an involution, a singular special Kaehler metric on the 2-sphere, or the unit interval. (2) Any complete hyperkaehler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.
In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such
spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.
