Existence of generalized totally umbilic 2-spheres in perturbed 3-spheres

TL;DR

This paper proves that small perturbations of the standard 3-sphere metric contain generalized totally umbilic 2-spheres, extending previous non-existence results and revealing new geometric structures via perturbative and analytical methods.

Contribution

It introduces a perturbative approach to establish the existence of generalized totally umbilic spheres in nearly round 3-spheres, including cases with degenerate traceless Ricci tensor.

Findings

Existence of uncountably many Willmore spheres on perturbed 3-spheres.

Generalized totally umbilic 2-spheres exist near the round metric.

Results apply to both analytic and smooth perturbations under certain conditions.

Abstract

It was recently shown by R. Souam and E. Toubiana that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics sufficiently close to the round metric $g_{0}$ on $\mathbb{S}^{3}$ possess \textsl{generalized} totally umbilic 2-spheres, namely critical points of the conformal Willmore functional $\int_{\Sigma}|A^\circ|^{2}\,d\mu_{\gamma}$. The same is true in the smooth setting provided a suitable non-degeneracy condition on the traceless Ricci tensor holds. The proof involves a gluing process of two different finite-dimensional reduction schemes, a sharp asymptotic analysis of the functional on perturbed umbilic spheres of small radius and a quantitative Schur-type Lemma in order to treat the cases when the traceless Ricci tensor of the perturbation is degenerate but not identically zero. For left-invariant metrics on $SU(2)\cong \mathbb{S}^{3}$ our result implies the existence of uncountably many distinct Willmore spheres.