Foliations of asymptotically flat manifolds by surfaces of Willmore type

2009/03/06 by Tobias Lamm, Lamm, Tobias, Jan Metzger +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.0903.1277

74 pages

arxiv created 2009/03/06 · openalex publication_date 2009/03/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Thus our result has applications in the theory of General Relativity.

Citations

• Minimal Varieties in Riemannian Manifolds
• The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation
• Constant curvature foliations on asymptotically hyperbolic spaces
• Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics
• Existence and Uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds

Cited by

• Foliations by critical surfaces of the Hawking energy in asymptotically flat initial data sets
• Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces
• Optimal constants of L² inequalities for closed nearly umbilical hypersurfaces in space forms

Related

• Local foliation of manifolds by surfaces of Willmore type
• The area preserving Willmore flow and local maximizers of the Hawking mass in asymptotically Schwarzschild manifolds
• Local foliations by critical surfaces of the Hawking energy and small sphere limit
• Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds
• Foliation by Constant Mean Curvature Spheres on Asymptotically Flat Manifolds