Small surfaces of Willmore type in Riemannian manifolds

Abstract

In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By \emph{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 \cite{Mondino:2008} that implies the non-existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.