Construction of blow-up sequences for the prescribed scalar curvature equation on S n. I. Uniform cancellation

Abstract

For n ≤ 6, using the Lyapunov-Schmidt reduction method, we describe how to construct (scalar curvature) functions on S n, 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 C n - 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. © 2012 World Scientific Publishing Company.