Upper bound for the total mean curvature of spin fill-ins

TL;DR

This paper proves Gromov's conjecture on upper bounds for the total mean curvature of boundary manifolds under certain conditions, extending to spin manifolds and specific boundary geometries with explicit constants.

Contribution

It establishes the conjecture for spin manifolds with bounds depending on mean curvature and provides explicit constants for boundaries in space forms, projective spaces, spheres, and flat tori.

Findings

Proves Gromov's conjecture for spin manifolds with mean curvature bounds.

Provides explicit constants for boundaries in space forms and special geometries.

Extends the conjecture to cases with negative mean curvature values.

Abstract

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.