A Fully Nonlinear Version of the Yamabe Problem

Abstract

Let (M, g_{0}) be a compact Riemannian manifold of dimension n ≥ 3, and let f be a symmetric function defined on ℝ^{n}. We are interested in finding some metric g in the conformal class of g_{0} such that f(λ(A_{g})) = 1, where A_{g} = \frac{1}{n−2}(Ric_{g} − \frac{R_{g}}{2(n−1)}g) is the Schouten tensor of g, and λ(A_{g}) denotes the eigenvalues of A_{g}. If the scalar curvature of the background metric of the manifold is positive, and f(λ) = ∑_{i} λ_{i}, this is the Yamabe problem. I will

also present the Liouville-type theorems for some conformally invariant fully nonlinear equations on ℝ^{n}. These are the joint works with Yanyan Li.