The Dirac operator under collapse with bounded curvature and diameter

Abstract

A sequence (<em>M_i</em>, <em>g_i</em>)_<em>i</em> of closed Riemannian manifolds with uniform bounded curvature and diameter collapses if it converges to a lower dimensional compact metric space (<em>B</em>,<em>h</em>). The limit space (<em>B</em>,<em>h</em>) has in general many singularities. <br /> In the first part of this thesis we show that the case, where the limit space (<em>B</em>,<em>h</em>) is at most of one dimension less, can be characterized by a uniform lower bound on the quotient of the volume of the manifolds<em>_i</em> divided by their injectivity radius. In that case the limit space (<em>B</em>,<em>h</em>) is a Riemannian orbifold. <br /> 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 (<em>B</em>,<em>h</em>). Lott showed that the spectrum of Dirac type operators converges to the spectrum of a certain first order elliptic differential operator <em> D </em> on <em>B</em>. We accentuate this result in the case of spin manifolds by giving an explicit description of the differential operator <em> D </em> and conclude that <em> D </em> is self-adjoint. Moreover we characterize the special case where <em> D </em> is the Dirac operator on <em>B</em>.