  We consider critical points of the global squared $L^2$-norms of the second
fundamental form and the mean curvature vector of isometric immersions into a
fixed background Riemannian manifold under deformations of the immersion. We
use the critical points of the former functional to define canonical
representatives of a given integer homology class of the background manifold.
We study the fibration ${\mathbb S}^3 \hookrightarrow Sp(2)\stackrel
{\pi_{\circ}}{\rightarrow} {\mathbb S}^7$ from this point of view, showing that
the fibers are the canonical generators of the $3$-integer homology of $Sp(2)$
when this Lie group is endowed with a suitable family of left invariant
metrics. Complex subvarieties in the standard $\mb{P}^n(\mb{C})$ are critical
points of each of the functionals, and are canonical representatives of their
homology classes. We use this result to provide a proof of Kronheimer-Wrowka's
theorem on the smallest genus representatives of the homology class of a curve
of degree $d$ in ${\mathbb C}{\mathbb P}^2$, and analyze also the canonical
representability of certain homology classes in the product of standard
$2$-spheres. Finally, we provide examples of background manifolds admitting
isotopically equivalent critical points in codimension one for the difference
of the two functionals mentioned, of different critical values, which are
Riemannian analogs of alternatives to compactification theories that has been
offered recently.
