For a flat torus boundary, the average boundary mean curvature of any fill-in with scalar curvature at least -n(n-1) is bounded by an explicit function of the shortest noncontractible circle length.
Rigidity phenomena involving scalar curvature
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Fill-Ins of Tori with Scalar Curvature Bounded from Below
For a flat torus boundary, the average boundary mean curvature of any fill-in with scalar curvature at least -n(n-1) is bounded by an explicit function of the shortest noncontractible circle length.