An
$n$dimensional
$\lambda $hypersurface
$X:M\to {\mathbb{R}}^{n+1}$
is the critical point of the weighted area functional
${\int}_{M}{e}^{\left[t\right]\frac{1}{4}X{}^{2}}\phantom{\rule{0.3em}{0ex}}d\mu $ for
weighted volumepreserving variations, which is also a generalization of the
selfshrinking solution of the mean curvature flow. We first prove that if the
${L}^{n}$norm of the second fundamental
form of the
$\lambda $hypersurface
$X:M\to {\mathbb{R}}^{n+1}$ with
$n\ge 3$ is less than an explicit
positive constant
$K\left(n,\lambda \right)$,
then
$M$
is a hyperplane. Secondly, we show that if the
${L}^{n}$norm of the tracefree
second fundamental form of
$M$
with
$n\ge 3$ is less than an
explicit positive constant
$D\left(n,\lambda \right)$
and the mean curvature is suitably bounded, then
$M$
is a hyperplane. We also obtain similar results for
$\lambda $surfaces
in
${\mathbb{R}}^{3}$ under
${L}^{4}$curvature
pinching conditions.
