Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin with a constant that also depends on a lower bound on the mean curvature H (which is allowed to take negative values). If the boundary is a (not necessarily convex) hypersurface in a space form of non-negative curvature, then the constant can be made explicit in terms of the mean curvature of this model embedding.
If the boundary has constant sectional curvature \(\kappa>0\) and is a projective space of dimension \(n\equiv 3 \mod 4\) or a sphere, then the constant can be expressed in terms of \(\kappa\). If the boundary is a flat torus, then the constant can be expressed in terms of lattice data.