In this talk we will focus on geometric properties of complete non-compact self-shrinkers for the mean curvature flow which are confined into some regions of the ambient Euclidean space.

Notably, we will obtain natural restrictions that force bounded complete self-shrinkers to be compact and we will observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. These results were inspired by a conjecture by H.D. Cao concerning the extrinsic polynomial volume growth of complete self-shrinkers.

This is a joint work with Stefano Pigola.