The volume growth of hyperkaehler manifolds of type A_{\infty}

Abstract

We study the volume growth of hyperkaehler manifolds of type $A_{\infty }$ constructed by Anderson-Kronheimer-LeBrun and Goto. These are noncompact complete 4-dimensional hyperkaehler manifolds of infinite topological type. These manifolds have the same topology but the hyperkaehler metrics are depends on the choice of parameters. By taking a certain parameter, we show that there exists a hyperkaehler manifold of type $A_{\infty }$ whose volume growth is r^a for each 3<a<4.