Gap and rigidity theorems of \lambda-hypersurfaces

We study \lambda -hypersurfaces that are critical points of a Gaussian weighted area functional \int _{\Sigma } e^{-\frac {\vert x\vert^2}{4}}dA for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete \lambda -hypersurfaces in terms of the norm of the second fundamental form \vert A\vert. Second, we show that in one dimension, the only smooth complete and embedded \lambda -hypersurfaces in \mathbb{R}^2 with \lambda \geq 0 are lines and round circles. Moreover, we establish a Bernstein-type theorem for \lambda -hypersurfaces which states that smooth \lambda -hypersurfaces that are entire graphs with polynomial volume growth are hyperplanes. All the results can be viewed as generalizations of results for self-shrinkers.