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

TL;DR

This paper investigates the volume growth of noncompact hyperkaehler manifolds of type A_infinity, demonstrating the existence of manifolds with volume growth rates between r^3 and r^4, depending on parameter choices.

Contribution

It constructs hyperkaehler manifolds of type A_infinity with adjustable volume growth rates, expanding understanding of their geometric properties.

Findings

Existence of hyperkaehler manifolds with volume growth r^a for 3<a<4.

Volume growth depends on parameter choices in the construction.

Manifolds share the same topology but differ in metric properties.

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.