XuJia Wang Australian National University
Fully nonlinear partial differential equations and applications in geometry
We discuss recent progresses on the kHessian equation and its conformal counterpart. The kHessian 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
kHessian 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 kHessian equation, namely the kYamabe 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.
