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.

References

A completely quasi-local proof of the non-negativity of the Wang--Yau, Brown--York, or Liu--Yau mass in full generality appears to be substantially more delicate than the corresponding argument obtained by asymptotically flat extension, and the problem remains open in this generality.

— Compact Proof of the Positivity of Quasi-Local Masses for a class of Initial Data  (2609.17361 - Mondal et al., 15 Sep 2026) in Section 1, subsection “Main Open Problem”

Gromov also conjectured p.~232 that the total mean curvature of a fill-in can be bounded from above in terms of the intrinsic boundary geometry and a lower scalar-curvature bound.

— Dirac eigenvalues and spacetime mean curvature estimates for DEC spin fill-ins  (2609.08720 - Raulot, 8 Sep 2026) in Section 1, Introduction, paragraph preceding Theorem 1.2