LAUSR

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 $$g^c$$gc, $$c\ge 0$$c≥0, on $$\mathbb {R}^4$$R4. The metric $$g^0$$g0 is the complex hyperbolic metric whereas the family $$(g^c)_{c>0}$$(gc)c>0 is equivalent to a family of metrics $$(h^b)_{b>0}$$(hb)b>0 depending on $$b=1/c$$b=1/c and smoothly extending to $$b=0$$b=0 for which $$h^0$$h0 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.