Gap theorems for complete self-shrinkers of rr-mean curvature flows

In this paper, we prove gap results for complete self-shrinkers of the $r$-mean curvature flow involving a modified second fundamental form. These results extend previous results for self-shrinkers of the mean curvature flow due to Cao-Li and Cheng-Peng. To prove our results we show that, under suitable curvature bounds, proper self-shrinkers are parabolic for a certain second-order differential operator which generalizes the drifted Laplacian and, even if is not proper, this differential operator satisfies an Omori-Yau type maximum principle.Comment: 22 pages. In this new version, the presentation and proof of Lemma 3.1 have been simplified, and some typos have been corrected. The main results remain unchange