  We study geometric properties of complete non-compact bounded self-shrinkers
and obtain natural restrictions that force these hypersurfaces to be compact.
Furthermore, we observe that, to a certain extent, complete self-shrinkers
intersect transversally a hyperplane through the origin. When such an
intersection is compact, we deduce spectral information on the natural drifted
Laplacian associated to the self-shrinker. These results go in the direction of
verifying the validity of a conjecture by H. D. Cao concerning the polynomial
volume growth of complete self-shrinkers. A finite strong maximum principle in
case the self-shrinker is confined into a cylindrical product is also
presented.
