Calderbank, D.M.J. and Singer, M.A. (2003) Einstein metrics and complex singularities. Inventiones Mathematicae, 156 (2). pp. 405-443. ISSN 1432-1297
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 hyperkähler 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 Kähler metric (which is hyperkähler if and only if KX is trivial), and that if KX is strictly nef, then X also admits a complete (non-Kähler) 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.
|Institution:||The University of York|
|Academic Units:||The University of York > Mathematics (York)|
|Depositing User:||York RAE Import|
|Date Deposited:||29 May 2009 11:41|
|Last Modified:||29 May 2009 11:41|
|Publisher:||Springer Verlag (Germany)|