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 ⊂ PN 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 ≥ 4, which if $n=4$ is very general and not of type $(2,2)$, does not carry any Ulrich bundles of rank r ≤ 3 unless $n=4, r=2$ and X is a quadric.