  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.
