Rigidity of complete self-shrinkers whose tangent planes omit a nonempty set

In this paper we prove rigidity results for the sphere, the plane and the right circular cylinder as the only self-shrinkers satisfying a new geometric assumption, namely the union of all tangent affine submanifolds of a complete self-shrinker omits a non-empty set of the Euclidean space. This assumption lead us to a new class of submanifolds, different from those with polynomial volume growth or the proper ones (which was proved to be equivalent by Cheng and Zhou, see Proc. Amer. Math. Soc. 141 (2013), no. 2, 687-696). In fact, in the last section, we present an example of a non proper surface whose tangent planes omit the interior of a right circular cylinder, which proves that these classes are distinct from each other.