The scalar curvature deformation equation on locally conformally flat manifolds of higher dimensions

Yu Yan

Vol. 237 (2008), No. 2, 373–398

DOI: 10.2140/pjm.2008.237.373

Abstract

We study the equation Δ_gu − (n − 2)∕(4(n − 1))R(g)u + Ku^p = 0 for p in 1 + ζ ≤ p ≤ (n + 2)∕(n − 2) on locally conformally flat compact manifolds (Mⁿ,g). We prove that when the scalar curvature R(g) ≡ 0 and n ≥ 5, under suitable conditions on K, all positive solutions u with bounded energy have uniform upper and lower bounds. In our previous 2007 paper, we also assumed this energy bound condition for the uniform estimates in the lower-dimensional case. We now give an example showing that this condition is necessary.

Keywords

scalar curvature, conformal deformation, uniform estimates

Mathematical Subject Classification 2000

Primary: 53C21

Milestones

Received: 24 January 2007

Revised: 6 May 2008

Accepted: 9 June 2008

Published: 1 October 2008