Some integral inequalities for L operator and their applications on self-shrinkers

In this paper, we derive a Reilly-type inequality for the drifting Laplacian operator L on a self-shrinker of the mean curvature flow. Using this Reilly-type inequality, we obtain some new Poincaré-type inequalities not only on the self-shrinker, but more interestingly, also on its boundary. Moreover, we get some eigenvalue estimates for L operator on the compact self-shrinker and its boundary, and also make some global estimates on generalized mean curvature Hρ on the boundary.