The $L^2$-norm of the second fundamental form of isometric immersions into a Riemannian manifold

TL;DR

This paper studies critical points of the squared $L^2$-norms of the second fundamental form for isometric immersions, revealing their role in defining canonical homology representatives and analyzing geometric structures in various manifolds.

Contribution

It introduces a novel variational approach to identify canonical homology representatives via critical points of the second fundamental form norm functional.

Findings

Fibers of the fibration ${ ext{S}}^3 o Sp(2) o { ext{S}}^7$ are canonical generators of homology.

Complex subvarieties are critical points and serve as canonical homology representatives.

Examples of critical points with different values provide insights into geometric and topological structures.

Abstract

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.