Elliptic and Parabolic Partial Differential Equations on Manifolds

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The principal mathematical objects we use to understand physical theories are partial differential equations (PDEs). There are many such equations, and the behavior of their solutions reflects the different kinds of phenomena we observe including the dispersion of heat, the effect of gravity, the motion of fluids and the movement of subatomic particles. Moreover, mathematicians use PDEs to understand geometric spaces and the possible structures that can exist on them. This project investigates the use of PDEs in two kinds of geometric problems. The first concerns the Calabi-Yau equation: this is a PDE used as a model in string theory and has wide-ranging applications in the study of geometric spaces defined by algebraic equations. A goal of this project is to generalize and solve the Calabi-Yau equation on spaces with much less structure, with a long term aim of classifying all such spaces. The second kind of geometric problem concerns a phenomenon known as collapsing. This occurs in the study of geometric heat equations where a geometric space evolves in time and may collapse in some directions to yield a lower dimensional object. This collapsing can reveal the structure of the original space. In order to carry out these investigations, the PI will need to develop new technical tools. The PI will take advantage of techniques which have been developed for classical equations such as the heat equation, and will adapt them to the study of non-linear PDEs occurring in geometry.This project will investigate nonlinear elliptic and parabolic equations, with applications to complex and almost geometry. In particular, the PI will study the problem of prescribing volume forms on manifolds, extending the well-known theorem of Yau for compact Kahler manifolds. Building on the PI's work on Hermitian and Gauduchon manifolds, the PI will investigate the question of prescribing volume forms for balanced metrics, and for almost Kahler metrics on four-manifolds. Another major goal of this project is to understand the phenomenon of collapsing along geometric flows. Collapsing for the Kahler-Ricci flow at infinite time is now quite well-understood. The PI will focus on the difficult problem of finite time collapse. This occurs for the Kahler-Ricci flow on Fano manifolds and also for the Chern-Ricci flow (a flow of Hermitian metrics) on non-Kahler complex manifolds such as the Hopf surface. To accomplish these goals, the PI will develop new tools for the study of nonlinear PDE. In particular, the PI will consider new second order estimates exploiting the convexity of the largest eigenvalue of Hessian. These kinds of estimates have already been used successfully to establish constant rank theorems for a general class of PDEs and optimal regularity results for the degenerate complex Monge-Ampere equation. The PI will also develop multi-point maximum principles, which have a long history in the study of convexity properties of solutions to PDEs, in the context of complex geometry.