Elliptic and parabolic PDEs on manifolds

OVERVIEW The PI plans to investigate elliptic and parabolic PDEs and geometry, under three broad themes. 1. Prescribing volume forms. Yau's Theorem states that one can prescribe the volume form of a Kahler metric on a compact Kahler manifold. This result is equivalent to an elliptic complex Monge-Ampere equation. The PI and his collaborators have extended Yau's Theorem to other settings, including Hermitian metrics and Gauduchon metrics. The PI will investigate the problem of prescribing the volume form of a balanced metric and will also study Donaldson's question of a version of Yau's theorem for symplectic 4-manifolds. 2. Geometric flows. The PI will study parabolic flows on manifolds, and in particular will investigate the phenomenon of collapsing in two geometric flows: the Kahler-Ricci flow and a flow of Hermitian metrics known as the Chern-Ricci flow. Collapsing at infinite time is now relatively well-understood, but there are many open questions concerning volume collapsing in finite time. In the case of the Chern-Ricci flow, this has possible applications to the study of Class VII surfaces. The PI will also investigate Donaldson?s geometric flow, a flow of symplectic forms on a 4-manifold which has possible applications to symplectic topology. 3. PDE methods. The PI will continue the investigation of two powerful and under-exploited PDE methods, in the context of problems in geometry and classical PDE. The first method concerns a maximum principle argument applied to the largest eigenvalue of the complex or real Hessian of a solution to a PDE, which has applications to PDE constant rank theorems and to complex geometry. The second method is to use multi-point functions and the maximum principle, extending work of B. Andrews, to study classical questions of convexity and also collapsing along parabolic PDEs in geometry. INTELLECTUAL MERIT Elliptic and parabolic PDEs comprise large classes of equations studied in physics and engineering, and also play fundamental roles in the study of geometry. The elliptic complex Monge-Ampere equation has far-reaching applications in Kahler geometry, including the existence of Ricci-flat metrics, the Kahlerity of K3 surfaces and the shape of the Kahler cone. The Kahler-Ricci flow is closely connected to the complex geometry of projective varieties, and it can be naturally extended to more general complex manifolds with a view towards classification questions. This project will develop tools to analyze these and other PDEs with the goal of proving new geometric results, while also shedding light on classical PDE problems. BROADER IMPACTS The PI will continue to be an active organizer of conferences, seminars and summer schools at Northwestern University and beyond. He will help to train graduate students at Northwestern and organize activities to engage students in the latest developments in the fields of PDE and geometry. The PI organizes a reading seminar at Northwestern with graduate students and postdoctoral scholars. The PI travels widely to give talks and lecture courses and proposes to continue doing this, to help to generate new ideas and train the next generation of mathematicians.