Huisken-Yau-type uniqueness for area-constrained Willmore spheres (Q6566420)

Let \((M, g)\) be \(C^4\)-asymptotic to Schwarzschild with positive mass. Suppose that the scalar curvature \(R\) satisfies \(R\geq -o(|x|^{-4} )\) as \(x\rightarrow \infty\). Then the authors show that there exist numbers \(\delta >0\), \(\lambda >1\) satisfying the following: there exists no area-constrained Willmore sphere \(\Sigma\) in \(M\) with nonnegative Hawking mass and\N\[\N\lambda (\Sigma )> \lambda , \quad \rho (\Sigma )<\delta \lambda (\Sigma ), \quad \log \lambda (\Sigma )<\delta \rho (\Sigma ),\N\]\Nwhere \(\lambda (\Sigma )\), \(\rho (\Sigma )\) are the area radius and the inner radius of \(\Sigma\) respectively.