  We prove a weak version of the Perdomo Conjecture, namely, there is a
positive constant $\delta(n)>0$ depending only on $n$ such that on any closed
embedded, non-totally geodesic, minimal hypersurface $M^n$ in
$\mathbb{S}^{n+1}$,
  $$\int_{M}S \geq \delta(n){\rm Vol}(M^n),$$ where $S$ is the squared length
of the second fundamental form of $M^n$. The Perdomo Conjecture asserts that
$\delta(n)=n$ which is still open in general. As byproducts, we also obtain
some integral inequalities and Simons-type pinching results on closed embedded
(or immersed) minimal hypersurfaces, with the first positive eigenvalue
$\lambda_1(M)$ of the Laplacian involved.
