Einstein metrics and complex singularities

D M J Calderbank, M A Singer

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkahler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kahler metric (which is hyperkahler if and only if K-X is trivial), and that if K-X is strictly nef, then X also admits a complete (non-Kahler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number.

Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable.

All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Original languageEnglish
Pages (from-to)405-443
Number of pages39
JournalInventiones mathematicae
Volume156
Issue number2
DOIs
Publication statusPublished - May 2004

Keywords / Materials (for Non-textual outputs)

  • SPACES
  • CONSTRUCTION
  • 4-MANIFOLDS
  • MANIFOLDS

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