  We consider critical points of the functionals $\Pi$ and $\Psi$ defined as
the global $L^2$-norm of the second fundamental form and mean curvature vector
of isometric immersions of compact Riemannian manifolds into a background
Riemannian manifold, respectively, as functionals over the space of
deformations of the immersion. We prove gap theorems for these functionals into
hyperbolic manifolds, and show that the celebrated gap theorem for minimal
immersions into the sphere can be cast as a theorem about critical points of
these functionals of constant mean curvature function, and whose second
fundamental form is suitably small in relation to it. In this case, the various
type of minimal submanifolds that can occur at the pointwise upper bound on the
norm of the second fundamental form are realized by manifolds of nonnegative
Ricci curvature, and of these, the Einstein ones are distinguished from the
others by being those that are immersed on the sphere as critical points of
$\Pi$.
