CONSTRUCTION OF BLOW-UP SEQUENCES FOR THE PRESCRIBED SCALAR CURVATURE EQUATION ON Sn I: UNIFORM CANCELLATION

Abstract

For n ≥ 6, using the Lyapunov–Schmidt reduction method, we describe how to construct (scalar curvature) functions on Sn, so that each of them enables the conformal scalar curvature equation to have an infinite number of positive solutions, which form a blow-up sequence. The prescribed scalar curvature function is shown to have Cn - 1,β smoothness. We present the argument in two parts. In this first part, we discuss the uniform cancellation property in the Lyapunov–Schmidt reduction method for the scalar curvature equation. We also explore relation between the Kazdan–Warner condition and the first-order derivatives of the reduced functional, and symmetry in the second-order derivatives of the reduced functional.