Large area-constrained Willmore surfaces in asymptotically Schwarzschild \(3\)-manifolds (Q6562505)

Let \((M, g)\) be an asymptotically flat Riemannian \(3\)-manifold with non-negative scalar curvature. Let \(\Sigma\) be a closed surface in \(M\) homeomorphic to the \(2\)-sphere. Then the Hawking mass of \(\Sigma\) is represented by the area and the Willmore energy of \(\Sigma\). In particular, \(\Sigma\) is an area-constrained stationary surface of the Hawking mass if and only if \(\Sigma\) is an area-constrained Willmore surface.\N\NSuppose that \((M, g)\) is a asymptotically Schwarzschild \(C^4\)-manifold with positive mass and satisfies additional asymptotic conditions on the scalar curvature. Using the main theorems in [\textit{T. Lamm} et al., Math. Ann. 350, 1--78 (2011; Zbl 1222.53028)], the authors show that the end of \(M\) is foliated by a family of on-center stable area-constrained Willmore spheres (Theorem 5), and they obtain conditions for area-constrained Willmore spheres to be leaves of this foliation (Theorem 8).