Geometry and Nonlinear Analysis

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low dimensional manifolds, particularly, 4-manifolds. The theory of Gromov-Witten invariants was established by using solutions of the Cauchy-Riemann equation (cf. [RT], [LT], [FO], [Si], [Ru]). These solutions are often refered as pseudo-holomorphic maps which are special minimal surfaces studied long in geometry. It is certainly not the end of applications of nonlinear partial differential equations to geometry. In this talk, we will discuss some recent progress on nonlinear partial differential equations in geometry. We will be selective, partly because of my own interest and partly because of recent applications of nonlinear equations. There are also talks in this ICM to cover some other topics of geometric analysis by R. Bartnik, B. Andrew, P. Li and X.X. Chen, etc. Standard partial differential equations in geometry are the Einstein equation, Yang-Mills equation, minimal surface equation as well as its close cousin, Harmonic map equation. There are also parabolic versions of these equations, leading to R. Hamilton’s Ricci flow, the Yang-Mills flow and the mean curvature flow. The solutions, which played a fundamental role in geometry and topology, of these equations are their self-dual type ones. I will focus on self-dual type solutions in this talk. All these equations are in general hyperbolic equations if we allow Lorentz metrics on the underlying manifolds, but in differential geometry, so far, we only concern static solutions, that is, we assume that the metrics involved are Riemannian. I do believe that the study of this static case will be very important in our future understanding general Einstein equation.