A fully nonlinear version of the Yamabe problem and a Harnack type inequality

Abstract

We present some results in [9], a continuation of our earlier works [7] and [8]. One result is the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds, and the other is a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.Let (M, g) be an n−dimensional, compact, smooth Riemannian manifold without boundary, n ≥ 3, consider the Weyl-Schouten tensor A g = 1 n−2

Application of the method of moving planes to conformally invariant equations

“…They tried to generalize it under some cumbersome conditions, see Remark 1. Recently, Li-Li [20] announced similar results of Theorem 2 and Theorem 3.…”

Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary

“…Our first existence result is for locally conformally flat manifolds with umbilic boundary. The existence of solutions of (1.8) with (f , ) satisfying (1.1)-(1.6) and f | ∂ = 0 has been proved in [27,29] on compact locally conformally flat manifolds without boundary. The proof of Theorem 1.1 is based on [27,29].…”

Liouville Theorem for Dunkl Polyharmonic Functions

“…Their starting point is the Almansi decomposition theorem for polyharmonic functions. We also refer to [10,11] for the extension of Liouville theorems for conformally invariant fully nonlinear equations. Recently, Gallardo and Godefroy [7] showed that if f is a bounded Dunkl harmonic function in R n , then it is a constant.…”