A spectral estimate for the Dirac operator on Riemannian flows

TL;DR

This paper provides a new upper bound for the smallest eigenvalues of the Dirac operator on Riemannian flows with transversal Killing spinors, with specific results on Sasakian and 3-dimensional manifolds, and compares it to existing lower bounds.

Contribution

It introduces a novel spectral estimate for the Dirac operator in the context of Riemannian flows, including classification results in special cases.

Findings

Derived an upper bound for eigenvalues on Sasakian and 3-manifolds.

Partially classified manifolds satisfying the limiting case.

Compared the new estimate with existing lower bounds.

Abstract

We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on Sasakian and on 3-dimensional manifolds and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.