  We study $\lambda$-hypersurfaces that are critical points of a Gaussian
weighted area functional $\int_{\Sigma} e^{-\frac{|x|^2}{4}}dA$ 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 $|A|$. 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.
