Yanyan Li
The Yamabe problem and the σk σ k -Yamabe problem
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 give in-depth study on the 𝜎𝑘-Yamabe problem and related topics. Emphasis will be given 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. An extension of the Yamabe problem is the so-called 𝜎𝑘-Yamabe problem, which is equivalent to solving certain fully nonlinear second order elliptic equations with certain conformal invariance structure. I will give a survey on results in the field, 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 among the following: Bernstein type arguments in making 𝐶1 and 𝐶2 priori estimates, Evans-Krylov estimates for second order uniformly fully nonlinear elliptic equations, theory of viscosity solutions, solving some 𝜎𝑘-Yamabe problem using parabolic flows.
If time permits, I will also present studies on the Nirenberg problem, the 𝜎𝑘-Nirenberg problem, and their corresponding parabolic flows. Open problems will be discussed.
There will be suggested papers
Math 517 or permission by instructor