The Hijazi inequalities on complete Riemannian Spin$^c$ manifolds

TL;DR

This paper extends Hijazi inequalities involving the Energy-Momentum tensor to eigenvalues of the Dirac operator on complete Riemannian Spin$^c$ manifolds, including the essential spectrum, under certain assumptions.

Contribution

It generalizes Hijazi inequalities to complete Spin$^c$ manifolds and explores the essential spectrum using refined inequalities and limiting case analysis.

Findings

Hijazi type inequality established for eigenvalues on complete Spin$^c$ manifolds

Extension of inequalities to the essential spectrum under additional assumptions

Analysis of limiting cases of the inequalities

Abstract

In this paper, we extend the Hijazi type inequality, involving the Energy-Momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spin$^c$ manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.