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

## TL;DR

This paper constructs a family of negatively curved quaternionic Kähler metrics that smoothly interpolate between real and complex hyperbolic geometries, revealing new geometric structures through quantum deformation.

## Contribution

It introduces a novel family of Einstein metrics interpolating between real and complex hyperbolic metrics via one-loop quantum deformation.

## Key findings

- The family includes the complex hyperbolic metric at one end.
- The family extends to the real hyperbolic metric at the other end.
- The conformal structure at infinity is explicitly determined.

## Abstract

We show that the one-loop quantum deformation of the universal hypermultiplet provides a family of complete $1/4$-pinched negatively curved quaternionic K\"ahler (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.