Low energy canonical immersions into hyperbolic manifolds and standard spheres

TL;DR

This paper studies critical points of geometric functionals related to the second fundamental form and mean curvature of isometric immersions, establishing gap theorems for hyperbolic manifolds and linking minimal immersions in spheres to these functionals.

Contribution

It introduces new gap theorems for these functionals in hyperbolic manifolds and connects classical minimal immersion results to critical points of these functionals.

Findings

Gap theorems for functionals into hyperbolic manifolds

Characterization of minimal immersions as critical points of these functionals

Distinction of Einstein manifolds among critical points

Abstract

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 $\Pi$.