Differential Equations in Geometry

在这项建议中，PI建议研究受物理启发的几何问题，特别是与弦理论和广义相对论相关的方程。在异质弦理论中，一组重要的非线性方程出现在复杂流形上的爱因斯坦和杨·米尔斯的耦合系统中。我们提出要解决这一问题。我们期望使用这个系统来理解高维流形的结构。Calabi-Yau流形上的复结构和高亏格曲线的模空间上的内蕴度量在代数几何和弦理论中起着非常重要的作用，我们期待着对它们的性质进行详细的研究。我们想把以前关于Rieman曲面的模空间的一些工作推广到更高维。在广义相对论中，我们将继续研究准局域质量的定义及其性质。在过去的四分之一个世纪里，理论物理和数学之间的相互作用取得了极其丰硕的成果。这项提议是这种互动的一部分。虽然这些工作将服务于物理学界，但我们相信，这项研究的结果将带来对几何的更好理解。

In this proposal, the PI proposes to study geometric problems that are inspired from physics, especially equations that are relevant to string theory and general relativity.An important set of nonlinear equations appeared in Heterotic string theory as a coupled system of Einstein and Yang Mills over a complex manifold. We propose to solve this system. We expect to use this system to understand structures of higher dimensional manifolds. Intrinsic metrics on moduli space of complex structures and curves of higher genus on Calabi-Yau manifolds play a very important role in algebraic geometry and string theory, we expect to study their properties in detail. We would like to generalize some of our previous works on moduli space of Rieman surfaces to higher dimension. In general relativity, we shall continue to study the definition of quasi-local mass and their properties.The interaction between theoretical physics and mathematics have been extremely fruitful in the past quarter of century. This proposal is part of this interaction. While the works will serve the physics community, we are confident that the outcome of this research will give insights that will lead to better understanding of geometry .