On a classification theorem for self-shrinkers

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the viewpoint of weighted manifolds and permits also to recover and to extend some others recent classification and gap results for self-shrinkers.