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 $B$.

37 pages, comments are welcome