复值解析挠率的渐近展开及其应用

Ray-Singer analytic torsion is an important object in global analysis of differential geometry. Recent years, studying the asymptotics of the Ray-Singer analytic torsion, especially the asymptotics on the hyperbolic manifolds, is a hot problem. These studies have some important results and these results have some applications in hyperbolic geometry, arithmetic geometry, topology and number theory. On the other hand, one has defined and studied the complex-valued analytic torsion. The complex-valued analytic torsion has some important relations with the Ray-Singer analytic torsion, for example on the odd dimensional manifolds the absolute value of the complex-valued analytic torsion equals the Ray-Singer analytic torsion. This project will study the asymptotics of the complex-valued analytic torsion and its applications, we will also study the noncommutative analytic torsion.

Ray-Singer解析挠率是整体微分几何的一个重要的研究内容。最近几年，研究Ray-Singer解析挠率在一些条件下的渐近展开，特别是在双曲流形上的渐近展开成为一个热点问题。这些研究的到了一些重要的结果，这些结果在双曲几何，算术几何，拓扑和数论等方面都有应用。另一方面，人们已经定义和研究了复值解析挠率。复值解析挠率和Ray-Singer解析挠率有很重要的联系，例如在奇数维流形上复值解析挠率的模等于Ray-Singer解析挠率。本项目计划研究复值解析挠率的渐近展开及其应用，我们还将对非交换的解析挠率进行研究。

解析挠率与正数量曲率是整体微分几何的重要研究内容。解析挠率是由Ray和Singer于1971年左右通过紧流形上的拉普拉斯算子的谱定义的。最近，Braverman-Kappeler，Burghelea-Haller，Cappell-Miller分别定义和研究了复值解析挠率。. 本项目主要在以下方面作了研究。1. 带边流形上的Cappell-Miller复值解析挠率的Cheeger-Mueller定理。2. Cappell-Miller全纯挠率的渐近展开公式。3. L^2解析挠率形式的正则性问题。4. 叶状流形上的正数量曲率与Lipschitz常数的关系。5. 解析挠率的粘合公式。