On Lorentzian Einstein affine hyperspheres

The classification of affine hyperspheres with Einstein affine metric is still an interesting unsolved problem in equiaffine differential geometry, although several nontrivial examples are well known. As the first attempt, the first author and his coauthors recently solved the problem for four-dimensional locally strongly convex case. In this paper, as another attempt, we study the problem in the Lorentzian case. As the main result, we completely classify four-dimensional affine hyperspheres with Lorentzian Einstein affine metric and non-vanishing Pick invariant.