the first non-round einstein metrics on spheres were described in 1973 by jensen in dimensions 4n+3 (n > 0). for the next 25 years it remained an open problem whether the same could be done in even dimensions. this question was settled in 1998 when c. böhm constructed infinite families of einstein metrics on all spheres of dimension between 5 and 9, in particular on $s^6$ and $s^8$.

in the 25 years since then, all spheres of odd dimension (at least 5) have been shown to admit non-round einstein metrics. however, there have been no new developments in even dimensions above 8, leaving open to speculation the question of whether, if the dimension is even, non-uniqueness of the round metric is a low-dimensional phenomenon or to be expected everywhere.

i will give an overview of the methods used to construct such einstein metrics, which we recently used to construct the first examples of non-round einstein metrics on $s^{10}$.

this is joint work with matthias wink.