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

Kota Hattori

doi:10.1007/s12220-010-9173-9

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

Kota Hattori

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	920-949
Number of pages	30
Journal	Journal of Geometric Analysis
Volume	21
Issue number	4
DOIs	https://doi.org/10.1007/s12220-010-9173-9
Publication status	Published - 2011 Oct
Externally published	Yes

Keywords

Gibbons-Hawking ansatz
Hyper-Kähler quotient
Volume growth

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.1007/s12220-010-9173-9

Cite this

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title = "The volume growth of hyper-K{\"a}hler manifolds of type A ∞",

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

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author = "Kota Hattori",

note = "Funding Information: Acknowledgements The author would like to thank Professor Hiroshi Konno for advice on this paper. The author was supported by Grant-in-Aid for JSPS Fellows (20 · 7215).",

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AB - 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.

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