Low density ferromagnetism in the Hubbard model

A single-band Hubbard model with nearest and next-nearest neighbour hopping is studied for $d=1$, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of $U$ for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M\"uller--Hartmann. For three dimensions and non-pathological band structures, it is proven that such a scenario does not work.