The Dirac operator under collapse to a smooth limit space

TL;DR

This paper explicitly describes the limit of Dirac operators on collapsing spin manifolds, showing their self-adjointness and characterizing when they coincide with the Dirac operator on the limit space.

Contribution

Provides an explicit description of the limiting Dirac operator under manifold collapse, extending previous spectral convergence results.

Findings

The limit operator is explicitly characterized.

The limit operator is self-adjoint.

Conditions when the limit operator is the Dirac operator on the base.

Abstract

Let $(M_i, g_i)_{i \in \mathbb{N}}$ be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower dimensional Riemannian manifold $(B,h)$ in the Gromov-Hausdorff topology. Lott showed that the spectrum converges to the spectrum of a certain first order elliptic differential operator $\mathcal{D}$ on $B$. In this article we give an explicit description of $\mathcal{D}^B$. We conclude that $\mathcal{D}^B$ is self-adjoint and characterize the special case where $\mathcal{D}^B$ is the Dirac operator on $B$.