Nonlinear PDEs and complex geometry

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Many of the laws of physics, such as the Einstein field equations, are described using nonlinear differential equations on geometric spaces. Geometric analysts use nonlinear differential equations to try to classify and understand the possible structures that can occur in geometric spaces. An equation known as the Ricci flow has been used to great success to show that every three dimensional space can be cut up into pieces, each of which has a well-understood geometry. The Ricci flow works by evolving a metric, which defines a notion of distance between two points, by a nonlinear heat flow process. However, for spaces of dimension four or higher (such as the space-time we live in), this process is not yet well understood. A major goal of this project is to investigate nonlinear equations on spaces of dimension four and higher with the aim of classifying these spaces and understanding the natural geometric structures that live on them. This project focuses on spaces endowed with a 'complex structure'. Despite the name, spaces with complex structures are easier to study, since this additional structure limits the kinds of singularities that could occur. Complex geometries appear in physical theories, such as string theory. This project will investigate complex geometries in four dimensions, using a new heat flow equation known as the Chern-Ricci flow. In higher dimensions, an equation analogous to the Calabi-Yau equation (much studied in string theory) will be used to investigate the structure of these spaces.This project will investigate nonlinear PDEs in complex geometry, with a focus on spaces which are non-Kahler. The PI will build on results for the Kahler-Ricci flow on complex surfaces to understand the behavior of the Chern-Ricci flow, a PDE which makes sense even in the non-Kahler case. This project will investigate the behavior of the Chern-Ricci flow in relation to the classification of complex surfaces. In particular, the PI will study how exceptional curves are contracted by this flow, and how the flow behaves on Class VII surfaces. For higher dimensional non-Kahler complex manifolds, the PI will use the complex Monge-Ampere equation to study questions about cohomology and existence of Kahler currents. In addition, the PI will study a new Monge-Ampere equation for (n-1)-plurisubharmonic functions, which is related to Gauduchon and balanced metrics on complex manifolds. A solution to this new equation will solve a long-standing conjecture of Gauduchon and have possible applications to deformation problems for projective varieties. Finally, the PI intends to extend these ideas to a Monge-Ampere type equation of Donaldson on symplectic 4-manifolds with compatible almost complex structures. A conjecture of Donaldson on existence of solutions to this equation can be reduced to a second order estimate. If this estimate holds it would give applications to the symplectic topology of 4-manifolds.