Math 517 or Instructor permission
In this course, We will discuss some partial differential equations arising from the study of conformally invariant quantities in Riemannian geometry.
A classical problem in this field is the Yamabe problem, that is to find on a compact Riemannian manifold a conformal metric which has constant scalar curvature. This is equivalent to solving a conformally invariant semi-linear partial different equation of critical exponent.
We will introduce the problem and give a review of the solution to this problem as well as results on compactness of the solution set. We will outline the proofs of these results.
We will then focus on a fully nonlinear version of the Yamabe problem, starting from an introduction of the problem and presenting up-to-date results on the problem. We will provide details of proofs to key results. To be more specific, we will present the following: Liouville theorems for conformally invariant elliptic and degenerate elliptic fully nonlinear equations in Euclidean equations, local gradient estimates and second derivatives estimates for the fully nonlinear Yamabe problem, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations and open problems, the existence and compactness results for the fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds, the existence results for the problem on general Riemannian manifolds for second elementary symmetric functions case making use of the variational structure of the problem, the
existence and compactness results for the problem for the \(k-\)elementary symmetric functions cases when \(k\) is not less than half of the dimension of the Riemannian manifolds (no available variational structure).
In the course, we will present a number of open problems which should be accessible to graduate students and younger researchers.