Asymptotic Analysis of embedded Willmore spheres in 3-dimensional manifolds

Abstract

In this paper, we show that, under arbitrary bounded Willmore energy assumption, embedded Willmore spheres (or more generally, embedded Willmore spheres under area constraint) with small diameter in a given $3$-dimensional Riemannian manifold $(M, h)$ necessarily concentrate at a critical point of the scalar curvature of $(M, h)$. This generalizes the result obtained by Laurain and Mondino for embedded Willmore spheres of small energy.