Small eigenvalues of the Dirac operator, Surgeries and Bordism Theory

狄拉克算子的小特征值、外科手术和 Bordism 理论

• 批准号: 75179442
• 负责人: Professor Dr. Bernd Eberhard Ammann
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2010-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/75179442?language=en
• 关键词: Small; eigenvalues; Dirac; operator; Surgeries

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.

在该项目中，我们想要研究有关狄拉克算子的小特征值及其与边函数理论的联系的几个主题。我们将研究以下主题： (1) 没有调和旋量的流形的 Bordism 理论：调和旋量是特征值 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 指数理论中的拓扑意义。我们希望扩展这些问题的最新进展，以获得没有调和旋量的流形的边主义理论。 (2) 等变调和L2-旋量。1 我们研究了在紧致流形的可能非紧覆盖上狄拉克算子的调和L2-旋量和Fredholmness。 (3) 使用手术技术规定狄拉克算子的小特征值。 (4)共形协变算子的ó型不变量及其在棱函数下的不变量。 (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属性的度量的存在方面发挥着核心作用。