Xu-Jia Wang

Fully non-linear partial differential equations and applications in geometry

We discuss recent progresses on the k-Hessian equation and its conformal counterpart. The k-Hessian equation is determined by the kth elementary symmetric polynomial of the eigenvalues of the Hessian matrix. It is the Laplace equation when k = 1 and a second order, fully nonlinear partial differential equation when k ≥ 2. The k-Hessian equation is of divergent form. We will introduce various variational properties of the equation, in particular a Sobolev type inequality. In conformal geometry we are concerned with the existence of solutions to the corresponding conformal k-Hessian equation, namely the k-Yamabe problem. It is the classical Yamabe problem when k = 1. We show the existence of solutions if either k ≥, or the equation is variational, which includes the cases when k = 1,2 or when the manifold is locally conformal flat. The existence has also been established for equations determined by more general symmetric functions.