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 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.