A lower bound for l-2 length of second fundamental form on minimal hypersurfaces

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 Sn+1, integral(M) S >= delta(n) 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.