Small eigenvalues of the Dirac operator, Surgeries and Bordism Theory

本课题主要研究狄拉克算子的小特征值及其与边界理论的关系。我们将研究以下主题：(1)无调和旋量流形的bordism理论：调和旋量是特征值为0的旋量。这个特征值起着非常特殊的作用，主要是由于它在Atiyah-Singer指标理论中的拓扑意义。为了得到无调和旋量流形的泛体理论，我们想推广这类问题的最新进展。(2)等变谐波l2旋量研究了紧流形可能非紧覆盖上的调和l2旋量和狄拉克算子的Fredholmness。(3)利用外科技术规定狄拉克算子的小特征值。(4) ó-type共形协变算子的不变性及其在边界下的不变性。(5)有边界的流形。(6)与广义相对论的关系。对于所有这些主题，边界和手术在证明具有某些性质的度量的存在性方面起着核心作用。

In the project we want to study several topics about small eigenvalues of the Dirac operator and its connections to bordism theory. We will study the following topics: (1) Bordism theory for manifolds without harmonic spinors: Harmonic spinors are spinors to the eigenvalue 0. This eigenvalue plays a very particular role, mainly due to its topological significance from Atiyah-Singer index theory. We want to extend recent progress on such questions in order to obtain a bordism theory for manifolds without harmonic spinors. (2) Equivariant harmonic L2-spinors.1 We study harmonic L2-spinors and Fredholmness of the Dirac operator on possibly non-compact coverings of compact manifolds. (3) Prescribing small eigenvalues of the Dirac operator using surgery techniques. (4) ó-type invariants of conformally covariant operators and its invariance under bordisms. (5) Manifolds with boundaries. (6) Relations to general relativity. For all these subjects bordisms and surgeries play a central role for proving the existence of metrics with certain properties.