The Yamabe problem and the 𝜎𝑘-Yamabe problem
A survey, and an exposition on methods and techniques in the study
In this course, We will discuss some partial differential equations arising from the study of conformally invariant quantities in Riemannian geometry. In particular we will survey results on the Yamabe problem and the 𝜎𝑘-Yamabe problem. We will also give an exposition on methods and techniques in the study. 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.
An extension of the Yamabe problem is the so-called 𝜎𝑘-Yamabe problem. I will survey results on the existence and compactness of solutions for the problem, and will discuss main open problems in the field.
The above will be in the format of a survey, with outlines of proofs. Embedded in the middle of such a survey, I will give more detailed discussions of a number of important methods and techniques involved in the study or closed related to the study of these problems. Such methods and techniques have played very significant roles in the development of the study of nonlinear partial differential equations. Such methods and techniques will be selected ampng the following: best Sobolev constant and extremal functions, symmetrizations and concentration compactness method, variational methods (lower semi-continuity and minimization in the Calculus of variations, the min-max method and the mountain pass theorem), Moser iterations and DeGorgi estimates, Liouville theorems for conformally invariant elliptic and degenerate elliptic fully nonlinear equations in Euclidean equations, the method of moving planes, small energy implies regularity, Bernstein type arguments in making 𝐶1 and 𝐶2 estimates, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations. theory of viscosity solutions.
No text book
Math 517 or Permission by instructor