On hyperkähler manifolds of type Ａ∞

Abstract

We study hyperkahler manifolds of type A∞ , which are noncompact real 4-dimensional manifolds of infinite topological type. Anderson-Kronheimer-LeBrun[1] and Goto [8] have constructed these hyperkahler manifolds whose metrics arecomplete and depend on the choice of parameters. We observe how the Riemannianmetrics and the complex structures which are induced from hyperkahler structuresdepend on the choice of parameters. In particular,· we focus on the asymptoticbehavior of these hyperkahler .metrics. By ·taking an appropriate parameter, weshow that there exists a complete hyperkahler manifold of type A∞ whose volumegrowth is rα for each 3 < α < 4.