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. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator $$\mathcal {D}^B$$ on B. We give an explicit description of $$\mathcal {D}^B$$ and characterize the special case where $$\mathcal {D}^B$$ equals the Dirac operator on B.