Refined position estimates for surfaces of Willmore type in Riemannian manifolds (Q6083958)

The Willmore energy is a conformally invariant energy of immersions of surfaces first introduced by \textit{S. D. Poisson} [Mémoires L'Inst. Sci. Arts 13, 167--225 (1814)] and \textit{S. Germain} [Mémoire sur cette question proposée par la première classe de l'Institut : Donner la théorie mathématique des vibrations des surfaces élastiques, et la comparer à l'expérience, Académie des Sciences (1815)]. The conformal invariance was discovered a century later by \textit{W. Blaschke} [Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. III: Differentialgeometrie der Kreise und Kugeln. Bearbeitet von \textit{G. Thomsen}. Springer, Cham (1929; JFM 55.0422.01)]. However, it was only thanks to work of \textit{T. J. Willmore} [An. Sti. Univ. Al. I. Cuza Iasi, N. Ser., Sect. Ia 11B, 493--496 (1965; Zbl 0171.20001)] and his celebrated conjecture about the minimisation of the Willmore energy on tori (see [\textit{F. C. Marques} and \textit{A. Neves}, Ann. Math. (2) 179, No. 2, 683--782 (2014; Zbl 1297.49079)] for more details on this problem) that this functional could finally enter into mainstream mathematics. First considered for immersions into Euclidean space, it admits a natural extension to Riemannian manifolds \((M^m,h)\), where it takes the form \begin{align*} W(\vec{\Phi})=\int_{\Sigma}\left(|\vec{H}|^2+K_h(\vec{\Phi}_{\ast}T\Sigma)\right)d\mathrm{vol}_{g} \end{align*} where \(\vec{\Phi}:\Sigma\rightarrow M^m\) is a smooth immersion, \(\vec{H}\) is its mean curvature vector, and \(K_h\) is the sectional curvature, while \(g=\phi^{\ast}h\) is the induced metric on the given surface \(\Sigma\). This is the right generalisation of the Willmore energy since this functional is also conformally invariant thanks to work of \textit{B.-Y. Chen} [Boll. Unione Mat. Ital., IV. Ser. 10, 380--385 (1974; Zbl 0321.53042)]. In this article, the author restricts to \(3\)-dimensional (mostly compact) manifolds, and instead considers the functional \begin{align*} \mathcal{W}(\vec{\Phi})=\int_{\Sigma}|\vec{H}|^2d\mathrm{vol}_{g}=\int_{\Sigma}H^2d\mathrm{vol}_{g}, \end{align*} where \(H\) is the (scalar) mean curvature. Although it fails to be conformally invariant, it is more suited to the problem treated in the article, where the author refines previous estimates for immersions whose area is small and whose Willmore energy is close enough to \(4\pi\). Although the Willmore inequality -- that states that the Willmore energy of Euclidean immersions is bounded from below by \(4\pi\) -- is not satisfied in Riemannian manifolds, adding the small area constraint insures that Willmore surfaces whose energy is close enough to \(4\pi\) look like round spheres. The author first proves that in manifolds of bounded geometry, there is a linear control of the gradient of the scalar curvature by the area of any area-constraint Willmore immersion (of small area and small \(\mathcal{W}-4\pi\)). Furthermore, the region enclosed by the given immersion is included in a geodesic ball of radius \(\dfrac{3}{4}\mathrm{diam}(\vec{\Phi}(\Sigma))\). Finally, assuming that the scalar curvature \(\mathrm{Scal}\) is a Morse function, the author shows that every area-constraint Willmore surfaces satisfying the above smallness condition encloses a single point of \(\mathrm{Scal}\)'s critical set. More precise estimates are obtained for the maximum locus of this scalar function.