Asymptotic Analysis of embedded Willmore spheres in 3-dimensional manifolds

TL;DR

This paper proves that small embedded Willmore spheres in 3D manifolds tend to concentrate at points where the scalar curvature is critical, extending previous results to more general energy conditions.

Contribution

It generalizes prior work by showing concentration behavior of Willmore spheres under bounded energy, not just small energy, in 3D manifolds.

Findings

Embedded Willmore spheres with small diameter concentrate at scalar curvature critical points.

The result extends previous small-energy concentration results to bounded energy scenarios.

Provides a deeper understanding of the geometric behavior of Willmore spheres in 3D 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.