Small Surfaces of Willmore Type in Riemannian Manifolds

In this paper, 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 Br(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 [9] that implies the non- existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.