On complete Ricci--flat Kähler manifolds

We study complete noncompact Ricci-flat Kähler manifolds. In the case of Euclidean volume growth, we prove that the underlying complex manifold is naturally quasi-projective and that the asymptotic cone is unique. Moreover, we establish no semistability at infinity and prove convergence to the asymptotic cone at a polynomial rate. In the case of sub-Euclidean volume growth, we construct complete Ricci-flat Kähler threefolds with bounded sectional curvature and finite topological type that admit no Kähler compactification.