On fill-ins with scalar curvature bounded from below and an inequality of Hijazi-Montiel-Rold\'an

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-Rold\'an for the first Dirac eigenvalue with a recent theorem of B\"ar. Moreover, we give an alternative proof of the Hijazi-Montiel-Rold\'an inequality based on the work of B\"ar and B\"ar-Ballmann.