Ulrich subvarieties and the non-existence of low rank Ulrich bundles on complete intersections

We characterize the existence of an Ulrich vector bundle on a variety $X\subset P^N$ in terms of the existence of a subvariety satisfying some precise conditions. Then we use this fact to prove that a complete intersection of dimension $n\ge 4$, which if $n=4$ is very general and not of type $(2,2)$, does not carry any Ulrich bundles of rank $r\le 3$ unless $n=4,r=2$ and $X$ is a quadric.