The volume growth of hyper-Kähler manifolds of type A _∞

We study the volume growth of hyper-Kähler 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-Kähler manifolds of infinite topological type. These manifolds have the same topology, but the hyper-Kähler metrics depend on the choice of parameters. By taking a certain parameter, we show that there exists a hyper-Kähler manifold of type A _∞ whose volume growth is r ^α for each 3<α<4.