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The Dirac operator under collapse with bounded curvature and diameter


Roos,  Saskia
Max Planck Institute for Mathematics, Max Planck Society;

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Roos, S. (2018). The Dirac operator under collapse with bounded curvature and diameter. PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.

Cite as: https://hdl.handle.net/21.11116/0000-0005-0FF8-4
A sequence (Mi, gi)i of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (B,h). The limit space (B,h) has in general many singularities.
In the first part of this thesis we show that the case, where the limit space (B,h) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifoldsi divided by their injectivity radius. In that case the limit space (B,h) is a Riemannian orbifold.
In the second part, we discuss the behavior of Dirac eigenvalues on a collapsing sequence of spin manifolds with bounded curvature and diameter converging to a lower dimensional Riemannian manifold (B,h). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator D on B. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator D and conclude that D is self-adjoint. Moreover we characterize the special case where D is the Dirac operator on B.