Huisken–Yau-type uniqueness for area-constrained Willmore spheres

Abstract

Let $(M,g)$ be a Riemannian three-manifold that is asymptotic to Schwarzschild. We study the existence of large area-constrained Willmore spheres $\mathrm{\Sigma }\subset M$ with nonnegative Hawking mass and inner radius ρ dominated by the area radius λ. If the scalar curvature of $(M,g)$ is nonnegative, we show that no such surfaces with $log\mathit{\lambda }\ll \mathit{\rho }$ exist. This answers a question of Huisken.