preprint

Inserted: 20 jan 2026

Year: 2025

Abstract:

We consider fill-ins of spin manifolds with scalar curvature bounded by $-n(n-1)$. Gromov proposed a conjecture relating the infimum of the mean curvature of such a fill-in to the hyperspherical radius. We observe that the inequality conjectured by Gromov follows by combining an inequality of Hijazi-Montiel-Roldán for the first Dirac eigenvalue with a recent theorem of Bär. Moreover, we give an alternative proof of the Hijazi-Montiel-Roldán inequality based on the work of Bär and Bär-Ballmann.