Differential Equations in Geometry

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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 .