Compactness and global estimates for the geometric Paneitz equation in high dimensions

Abstract

Given ( M , g ) (M,g) , a smooth compact Riemannian manifold of dimension n ≥ 5 n \ge 5 , we investigate compactness for the fourth order geometric equation P g u = u 2 ♯ − 1 P_gu = u^{2^\sharp -1} , where P g P_g is the Paneitz operator, and 2 ♯ = 2 n / ( n − 4 ) 2^\sharp = 2n/(n-4) is critical from the Sobolev viewpoint. We prove that the equation is compact when the Paneitz operator is of strong positive type.