The Dirac operator under collapse to a smooth limit space

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 $B$.