Gap and rigidity theorems of $\lambda$-hypersurfaces

We study $\lambda$-hypersurfaces that are critical points of a Gaussian weighted area functional ${\int }_{\mathrm{\Sigma }}{e}^{-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{4}}dA$ 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 $\mathrm{\mid }A\mathrm{\mid }$. Second, we show that in one dimension, the only smooth complete and embedded $\lambda$-hypersurfaces in ${\mathbb{R}}^2$ with $\lambda \ge 0$ 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.