Einstein metrics and complex singularities

• 成果类型: Article
• 署名作者: Calderbank, DMJ; Singer, MA
• 署名单位: University of Edinburgh
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-003-0344-1
• 发表日期: 2004
• 页码: 405-443
• 关键词: spaces; CONSTRUCTION; 4-manifolds; MANIFOLDS

摘要

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.