Differential Equations in Geometry

在这项提议中，PI提议研究受物理学启发的几何问题，特别是与弦理论和广义相对论相关的方程。作为复流形上爱因斯坦和杨米尔斯的耦合系统，异质弦理论中出现了一组重要的非线性方程。我们建议解决这个系统。我们期望用这个系统来理解高维流形的结构。复杂结构模空间上的固有度量和Calabi-Yau流形上的高属曲线在代数几何和弦理论中有着非常重要的作用，我们期望对它们的性质进行详细的研究。我们想把以前关于黎曼曲面模空间的一些研究推广到高维。在广义相对论中，我们将继续研究准局部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卓有成效。这个提议是这种互动的一部分。虽然这些工作将为物理界服务，但我们相信，这项研究的结果将提供见解，从而更好地理解几何。

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 .