The Volume Growth of Hyper-Kähler Manifolds of Type A ∞

Abstract

We study the volume growth of hyper-Kahler manifolds of type A∞ constructed by Anderson–Kronheimer–LeBrun (Commun. Math. Phys. 125:637–642, 1989) and Goto (Geom. Funct. Anal. 4(4):424–454, 1994). These are noncompact complete 4-dimensional hyper-Kahler manifolds of infinite topological type. These manifolds have the same topology, but the hyper-Kahler metrics depend on the choice of parameters. By taking a certain parameter, we show that there exists a hyper-Kahler manifold of type A∞ whose volume growth is rα for each 3<α<4.