A construction of Poincar\'e-Einstein metrics of cohomogeneity one on the ball

TL;DR

This paper constructs a smooth family of Poincaré-Einstein metrics on the unit ball with Berger sphere infinities, using a Gibbons-Hawking-type ansatz, revealing new geometric structures and degenerations.

Contribution

It provides an explicit one-parameter family of Poincaré-Einstein metrics on the ball with novel boundary conformal structures, based on a Gibbons-Hawking-type ansatz.

Findings

Includes the hyperbolic metric as a special case.

Converges to complex hyperbolic metric at one end.

Degenerates to a lower-dimensional manifold with a conical singularity.

Abstract

We exhibit an explicit one-parameter smooth family of Poincar\'e-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ans\"atz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincar\'e-Einstein manifold of one lower dimension with an isolated conical singularity.