Nonlinear Partial Differential Equations and Geometry

Northwestern University, Evanston IL

Investigators

Abstract

Many physical theories have been modeled successfully by mathematical equations known as partial differential equations (PDEs). The classical examples are the heat equation, the wave equation and the Laplace equation, which describe in ideal conditions the behavior of heat, waves and electrostatic potentials respectively. These are linear PDEs. More complicated theories are often modeled by nonlinear PDEs. A classic nonlinear PDE is the Monge-Ampere equation which has connections to the physical theory of optics but is also a powerful tool in the study of geometry. This research will investigate several nonlinear PDE, all related to the Monge-Ampere equation, which arise in geometry or which exhibit geometric behavior. The research goals are two-fold: to use nonlinear PDEs to advance our understanding of geometric spaces and the structures that live on them; and to understand the phenomena and behavior of solutions of nonlinear PDE using the tools and language of geometry. In addition the PI will also train graduate students in the methods of nonlinear PDEs and geometry, and guide their research. The PI will carry out research on nonlinear PDEs and geometry. There are five projects, linked by the common theme of the complex Monge-Ampere equation and focusing on developing new analytic tools and strategies to derive a priori estimates. For the first project, the PI will investigate a conjecture of Donaldson extending Yau’s theorem on the complex Monge-Ampere equation to the setting of symplectic 4-manifolds, using an ansatz which reduces the nonlinear PDE to an equation of a single real-valued function. A second project will study Perelman’s estimates for the Kahler-Ricci flow with the goal of understanding the behavior of the flow when there is a finite time singularity. A further project is to investigate a non-Kahler version of this flow, known as the Chern-Ricci flow, with a focus on finite time singularities on complex surfaces. Another project will extend the work of Chen-Cheng and Shen on the question of existence of metrics with constant Chern scalar curvature, in the setting of non-Kahler complex surfaces. A final project is to study the uniform convexity of convex solutions to PDEs satisfying certain structure conditions, building on constant rank theorems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.