Geometric Applications of Dirac Operator and Atiyah-Singer Index Theory

项目编号：dms -1007041项目负责人：戴先哲。本文讨论了几何中与Dirac算子和Atiyah-Singer指数理论相关的几个问题。它包括狄拉克算子和旋量在研究爱因斯坦度量稳定性问题中的应用。PI将通过探索与kill旋量和sasaki -Einstein度规的联系，更好地理解具有正标量曲率的爱因斯坦度规的稳定性。另一个问题是利用局部指标理论技术研究热核和Bergman核及其与规范度量的关系。PI想研究Dirac算子的谱隙和半正情况下Bergman核的渐近展开。PI还将研究一类非紧流形上的Ricci流，即ALE空间。长期趋同的问题将是主要焦点。最后，PI将研究几何不变量在度量退化下的行为，包括绝热极限和圆锥退化。特别地，其中一个应用将是具有圆锥奇点的流形的Ray-Singer猜想。爱因斯坦的广义相对论将引力几何化，引力是自然界四种基本力之一，也是塑造我们宇宙的主导力量。爱因斯坦流形在数学和物理学中起着重要的作用。理解爱因斯坦流形的稳定性是很重要的。稳定性问题在里奇流等几何演化方程的研究中也很重要。最近的发展显示了利玛窦流的非凡力量。狄拉克算子及其相关的几何不变量在数学和物理的各个领域中发挥着重要的作用。这一建议旨在更好地理解爱因斯坦流形的稳定性，非紧化流形上的里奇流，以及几何不变量。

AbstractAward: DMS-1007041Principal Investigator: Xianzhe DaiThis proposal concerns several problems in geometry that are related to Dirac operators and Atiyah-Singer index theory. It includes the use of Dirac operator and spinors in the study of stability problem for Einstein metrics. The PI will seek better understanding of the stability of Einstein metrics with positive scalar curvature by exploring the connection with Killing spinors and Sasakian-Einstein metrics. Another problem involves the study of heat kernel and Bergman kernel using local index theory technique and the study of their relation with canonical metrics. The PI would like to study the spectral gap of Dirac operators and the asymptotic expansion of Bergman kernel in the semipostive case. The PI will also study Ricci flows on a class of noncompact manifolds, the ALE spaces. The question of long time convergence will be the main focus. Finally, the PI will investigate the behavior of geometric invariants under metric degeneration, including adiabatic limit and conical degeneration. In particular, one of the applications will be the Ray-Singer conjecture for manifolds with conical singularities.Einstein's General relativity geometrizes gravity, one of the four fundamental forces in nature and the dominating one in shaping our universe. Einstein manifolds play essential role in mathematics and physics. It is important to understand the stability of Einstein manifolds. Stability issue is also important in the study of geometric evolution equations such as the Ricci flow. Recent development shows the extraordinary power of the Ricci flow. Dirac operators and related geometric invariants are playing significant and important role in diverse fields of mathematics and physics. This proposal aims for better understanding of the stability of Einstein manifolds, of the Ricci flow on noncompact manifolds, and of geometric invariants.