Nonlinear Partial Differential Equations and Geometry

Overview The PI plans to carry out research on nonlinear PDEs and geometry. There are five projects in this proposal, linked by the common theme of the complex Monge-Ampere equation. 1. A symplectic Calabi-Yau theorem. This project is on a conjecture of Donaldson extending Yau's theorem on the complex Monge-Ampere equation to the setting of symplectic 4-manifolds. The PI will investigate a particular ansatz which reduces the nonlinear PDE to an equation of a single real-valued function. 2. Finite time singularities for the Kahler-Ricci flow. The Kahler-Ricci flow is a parabolic complex Monge-Ampere equation. The PI will revisit Perelman's estimates for the flow in the case of Fano manifolds with the goal of understanding more generally the behavior of the flow when there is a finite time singularity. 3. The Chern-Ricci flow and the Continuity equation. The PI will investigate the behavior of the Chern-Ricci flow, a non-Kahler version of the Kahler-Ricci flow, and its elliptic analogue, the Continuity equation. The focus of this project will be on finite time singularities on complex surfaces. 4. Hermitian metrics with constant Chern scalar curvature. Extending recent work of Chen-Cheng and X.S. Shen, this project will probe the question of existence of non-Kahler Hermitian metrics whose Chern scalar curvature is constant, with a special focus on the case of complex surfaces. 5. Uniform convexity of solutions to PDEs. The PI will study the question of uniform convexity of convex solutions to PDEs satisfying certain structure conditions, building on constant rank theorems. Examples include semi-linear equations and the complex Monge-Ampere equation on toric Fano manifolds. Intellectual Merit Partial differential equations are ubiquitous in geometry and the modeling of physical phenomena. This project will further our understanding of solutions to nonlinear PDEs which either arise naturally in geometry or which exhibit geometric behavior. In particular this project will use PDEs to shed light on the structures of symplectic manifolds with nonintegrable almost complex structures and non-Kahler complex manifolds, and the geometric properties of elliptic and degenerate parabolic PDEs. The focus of the project will be on developing new analytic tools and strategies to derive a priori estimates for a broad range of PDEs. Broader Impacts The PI will continue to organize local seminars at Northwestern University and play an active role in the training of graduate students and postdocs. The PI is advising one PhD student who plans to graduate in 2021 and expects to take on further students in the upcoming years. The PI will also supervise summer research projects for underrepresented minority students. In addition, the PI plans to organize national and international conferences in the field.