Eigenvalue pinching on $\text{spin}^c$ manifolds

TL;DR

This paper establishes eigenvalue pinching results for Dirac operators on $ ext{spin}^c$ manifolds, linking geometric convergence and the existence of Killing spinors, and analyzing metric and curvature regularity.

Contribution

It introduces a new notion of convergence for $ ext{spin}^c$ manifolds and studies the relationship between metric regularity and bundle curvature in the context of Killing spinors.

Findings

Pinching results for small Dirac eigenvalues on $ ext{spin}^c$ manifolds.

A new convergence framework for $ ext{spin}^c$ manifolds involving principal $ ext{S}^1$-bundles.

Analysis of metric and curvature regularity relations in $ ext{spin}^c$ manifolds with Killing spinors.

Abstract

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a principal $\mathbb{S}^1$-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal $\mathbb{S}^1$-bundle on $\text{spin}^c$ manifolds with Killing spinors.