Quarter-pinched Einstein metrics interpolating between real and complex hyperbolic metrics

Abstract

We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete $1/4$-pinched negatively curved quaternionic Kähler (i.e. half conformally flat Einstein) metrics $g^c$, $c\ge 0$, on ${\mathbb{R}}^4$. The metric $g^0$ is the complex hyperbolic metric whereas the family $(g^c{)}_{c>0}$ is equivalent to a family of metrics $(h^b{)}_{b>0}$ depending on $b=1/c$ and smoothly extending to $b=0$ for which $h^0$ is the real hyperbolic metric. In this sense the one-loop deformation interpolates between the real and the complex hyperbolic metrics. We also determine the (singular) conformal structure at infinity for the above families.

14 pages, accepted for publication in Mathematische Zeitschrift