Abstract
We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C^3-close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such a region is foliated by spheres of Willmore type, see (Lamm et al. in Math Ann 350(1):1–78, 2011). In this paper, we prove that the leaves of this foliation are stable under small area preserving W^{2,2}-perturbations with respect to the area preserving Willmore flow. This implies, in particular, that the leaves are strict local area preserving maximizers of the Hawking mass with respect to the W^{2,2}-topology.
Highlights
Let (M, g) be an asymptotically flat Riemannian three-manifold with non-negative scalar curvature
On the other hand, finding the right notion of quasi-local mass corresponding to this global invariant remains an interesting open problem
With the help of the Hawking mass, the ADM mass can be quantified in terms of the local geometry: in a celebrated work, Huisken and Illmanen used a weak version of the inverse mean curvature flow to prove the Riemannian Penrose inequality which states that the ADM mass of an asymptotically flat manifold is bounded from below by the Hawking mass of any connected outward minimizing surface
Summary
Let (M, g) be an asymptotically flat Riemannian three-manifold with non-negative scalar curvature. Given the results obtained for the isoperimetric problem, one is tempted to believe that in an asymptotic region, the leaves λ are the global maximizers of the Hawking mass and perhaps the only surfaces of Willmore type with non-negative Hawking mass and a sufficiently large area. Up to now, this has not even been known in Schwarzschild. For the convenience of the reader, we have included a summary of the argument used by Jachan in [19] in the appendix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
