The Geometry of PDEs

Institute: IAS     August 2009

leaf dropIn the twentieth century, many of the advances in the theory of patial differential equations were the result of studying specific geometric problems; for example, minimal surfaces. The program in Geometric Partial Differential Equations demonstrated that this interaction between geometry and PDE continues to flourish and to illuminate the connections between seemingly unrelated problems.

Conformal geometry is a subject with classical origins: complex analysis and the study of angle-preserving maps between planar domains. Studying the behavior of Riemannian manifolds under conformal deformations arises in both geometric and physical applications, including General Relativity and, more recently, the AdS-CFT correspondence in string theory.

A central problem in conformal geometry is the study of conformal invariants; that is, quantities which remain invariant when the underlying Riemannian manifold is deformed conformally. A general principle for constructing local conformal invariants has been developed by Fefferman and Graham, and involves the study of Poincaré-Einstein metrics. These are complete Einstein manifolds which are asymptotically hyperbolic and admit conformal compactifications. Applications of this theory have included the construction of the conformally invariant operators and associated curvature invariants, including the Q-curvature and σk-curvature, which can be viewed as various generalizations of the scalar curvature.

The PDE aspects of the theory of these invariants are examples of the more general problem of uniformizing a conformal manifold; that is, finding the “best” metric in a fixed conformal class. In the case of surfaces, “best” in this context usually means “of constant curvature” but in higher dimensions the curvature is algebraically more complicated, and there are various notions of uniformizing a conformal class. In the case of finding a conformal metric with constant σk-curvature, this leads to a fully nonlinear equation whose ellipticity condition amounts to a kind of “convexity” or “pinching” condition for the Ricci tensor. Although considerable progress has been made in understanding the exitence and regularity properties of these equations when the underlying manifold has positive curvature (in a suitable sense), the case of negative curvature exhibits entirely different behavior. Remarkably, the loss of regularity for these equations on negatively curved manifolds has been studied in a very different context, the theory of optimal transportation.

Optimal transportation can be traced to a monograph of Gaspard Monge, Sur la Théorie des Déblais et des Remblais, in which he proposed the problem of finding the optimal way of transporting a pile of earth. Although very geometric in its origins, the problem has applications to diverse fields, including economics and the design of reflector antennas. The last ten years has seen considerable progress in the existence and regularity theory, including the introduction of the Ma-Trudinger-Wang tensor that is now central to the PDE aspects of the subject. The connection with conformal geometry emerges for a certain choice of the “cost function” for the transportation problem, and the Monge-Ampere equation corresponding to an optimal map is precisely the σk-curvature equation for a negatively curved manifold; moreover, local regularity fails precisely because the MTW curvature fails to be positive. Although the structural reasons for the failure of interior regularity are well understood, it is an important open problem to give a geometric description of this phenomenon.

Other topics of activity during the year were free boundary problems and homogenization, which arise in image analysis and the study of phase transitions. Solutions of differential equations describing very irregular media often exhibit very complicated behavior, but are more regular when viewed on larger scales. The process of analyzing this regularizing effect is called homogenization, and appears in many PDE problems. For example, to study the formation of a drop of liquid on a surface, one needs to consider a variational problem with two energy terms, one of which reflects the inhomogeneities of the surface on small scales. These problems are very geometric in their formulation, but again require a fundamental understanding of the underlying PDEs.