Gap theorems for complete self-shrinkers of $r$-mean curvature flows

Abstract

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.

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 unchanged