Liouville theorem for fully nonlinear equations

Let (M, g) be an n-dimensional, smooth, compact Riemannian manifold without boundary. For n ≥ 3, the σ_k Yabame problem consists in finding metrics with constant σ_k curvature in the same conformal class of g. When k=1, it is known as the Yamabe conjecture, which states that there exist Riemannian metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture was sovled through the works of Yamabe, Trudinger, Aubin, and Schoen.

In this talk, we will focus on the fully nonlinear equations that arise naturally from the σ_k Yamabe problem. We obtain Schoen's Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. By the asymptotic behavior and the method of moving spheres, we give another proof of the Liouville theorem proved by A. Li and Y.Y. Li (2005). We also provide some applications of the method of moving spheres and moving plane method.