On area-constrained critical surfaces of the Hawking functional, zero Hawking energy implies the enclosed region is flat or the reference space form, but the dynamical statement is conditional on a restrictive technical assumption.
Modified Hawking mass and rigidity of three-manifolds with boundary
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abstract
In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a $3$-dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable two-disks. Finally, we show that the $3$-dimensional Riemannian manifold with boundary is locally isometric to the half anti-de Sitter-Schwarzschild manifold.
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Rigidity and positivity of Hawking quasi-local energy on area-constrained critical surfaces
On area-constrained critical surfaces of the Hawking functional, zero Hawking energy implies the enclosed region is flat or the reference space form, but the dynamical statement is conditional on a restrictive technical assumption.