Eigenvalue Estimates For The Dirac Operator On Kaehler-Einstein Manifolds Of Even Complex Dimension

TL;DR

This paper derives an improved lower bound for the first eigenvalue of the Dirac operator on Kähler-Einstein manifolds with positive scalar curvature and even complex dimension, demonstrating its sharpness in certain cases.

Contribution

It introduces a new lower bound for the Dirac operator's first eigenvalue in specific Kähler-Einstein manifolds and shows this bound can be attained by some manifolds.

Findings

New lower bound for eigenvalues established

Existence of manifolds achieving this bound demonstrated

Enhanced understanding of Dirac operator spectra on Kähler-Einstein manifolds

Abstract

In K\"ahler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.