Small surfaces of Willmore type in Riemannian manifolds. (English) Zbl 1202.53056

Lamm, Tobias; Metzger, Jan

Int. Math. Res. Not. 2010, No. 19, 3786-3813 (2010).

Summary: We investigate the properties of small surfaces of Willmore type in three-dimensional Riemannian manifolds. By small surfaces, we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball \(B_r(p)\) for arbitrarily small radius \(r\) around a point \(p\) in the Riemannian manifold, then the scalar curvature must have a critical point at \(p\). As a byproduct of our estimates, we obtain a strengthened version of the non-existence result of Mondino (to appear) that implies the non-existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.

MSC:

53C40

Global submanifolds

Keywords:

topological spheres; geodesic ball; Willmore functional; scalar curvature