The Hijazi inequality on conformally parabolic manifolds

TL;DR

This paper establishes the Hijazi inequality for Dirac eigenvalues on complete, finite-volume manifolds, extending it to the essential spectrum under certain conditions, and applies it to conformally parabolic manifolds with positive spin Yamabe invariant.

Contribution

It proves the Hijazi inequality in new geometric settings, including complete manifolds of finite volume and conformally parabolic manifolds, with conditions on scalar curvature and spectral elements.

Findings

Hijazi inequality holds for complete finite-volume manifolds.

Extension of the inequality to the essential spectrum under specific assumptions.

Conformal version of the Hijazi inequality for conformally parabolic manifolds with positive spin Yamabe invariant.

Abstract

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.