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 $(x,y,z)\mapsto (\lambda x,\lambda y,{\lambda }^{2}z)$, and we explain why it is the generator of these dilations, and not the Euclidean position vector, which appears in the self-shrinker equation. Our main result is that a complete properly immersed Legendrian self-shrinker with bounded Legendrian angle is a Legendrian $n$-plane. On a Legendrian self-shrinker the Legendrian angle is a constant multiple of the coordinate $z$, so that this is a statement about the position of the submanifold, and it has two consequences. A complete properly immersed Legendrian self-shrinker which lies in a slab $|z|\le C$ is a Legendrian $n$-plane, and in particular there are no closed Legendrian self-shrinkers in ${\mathbb{R}}^{2n+1}$; this has no counterpart for Lagrangian self-shrinkers in ${\mathbb{C}}^n$, among which there are closed ones. And the potential of an entire smooth Legendrian self-shrinking graph is a quadratic polynomial, without any further assumption. The proof uses a volume growth estimate for Legendrian self-shrinkers with respect to a Gaussian weight which is built from the horizontal part of the generator of the dilations.