Four-dimensional Einstein metrics from biconformal deformations

Abstract

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${\mathbb R}^2$.