Four-dimensional Einstein metrics from biconformal deformations

TL;DR

This paper develops methods to analyze biconformal deformations on four-manifolds with conformal foliations, enabling the construction of Einstein metrics and exploring their geometric properties, including asymptotic behaviors.

Contribution

It introduces tools for calculating Ricci curvature transformations under biconformal deformations and constructs explicit Einstein 4-manifolds using these methods.

Findings

Constructed new Einstein 4-manifolds via biconformal deformations.

Identified a family of examples with ends collapsing to R^2.

Provided formulas for Ricci curvature transformation under 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 ${\mathbb R}^2$.