Gromov’s total mean-curvature conjecture for fill-ins

Establish an upper bound for the total mean curvature of a fill-in determined solely by the intrinsic geometry of its boundary and a prescribed lower bound for the scalar curvature.

Background

The paper recalls a conjecture of Gromov concerning Riemannian fill-ins: the integral of the boundary mean curvature should admit an upper bound depending only on intrinsic boundary geometry and a lower scalar-curvature bound. The paper notes that Bär subsequently proved the corresponding estimate for spin fill-ins, with an additional dependence on a lower bound for the boundary mean curvature.

The authors combine Bär’s argument with a Jang-equation deformation to obtain a spacetime analogue in which the ordinary mean curvature is replaced by the norm of the spacetime mean-curvature vector. Thus, the recalled conjecture serves as the motivation and geometric precedent for Theorem 1.2, although the specific spin version cited in the paper has already been proved.