Rigidity theorems for complete Legendrian self-shrinkers

We study the self-shrinkers of the Legendrian mean curvature flow in the contact Euclidean space $\mathbb R^{2n+1}$, that is, the Legendrian submanifolds which move under the flow by the contact dilations $(x,y,z)\mapsto(λx,λy,λ^{2}z)$, and we explain why it is the generator of these dilations, and