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

TL;DR

This paper explores fill-ins of spin manifolds with scalar curvature bounds, connecting Gromov's conjecture on mean curvature to eigenvalue inequalities, and offers new proofs and insights into these geometric inequalities.

Contribution

It links Gromov's fill-in conjecture to eigenvalue inequalities and provides an alternative proof of the Hijazi-Montiel-Roldán inequality.

Findings

Gromov's conjecture follows from eigenvalue inequalities and Bär's theorem.

An alternative proof of the Hijazi-Montiel-Roldán inequality is provided.

Connections between scalar curvature bounds, mean curvature, and Dirac eigenvalues are established.

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.