A variational problem for submanifolds in a sphere

Let x: M -> Sn+p be an n-dimensional submanifold in an (n + p)-dimensional unit sphere Sn+p, M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional integral(M)(S - nH)(n/2)dv, where S = Sigma(alpha,i,j)(h(ij)(alpha))(2) is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons' type for n-dimensional compact Willmore submanifolds in Sn+p. In this paper, we discover that a similar integral inequality of Simons' type still holds for the critical submanifolds of the functional integral(M) (S - nH(2))dv. Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.