Refined estimates for simple blow-ups of the scalar curvature equation on S^n

TL;DR

This paper refines the analysis of blow-up behavior for the scalar curvature equation on spheres, providing more precise estimates by incorporating local curvature effects, global interactions, and balance formulas.

Contribution

It introduces a second order blow-up analysis for the conformal scalar curvature equation on S^n, enhancing the understanding of solution behavior near blow-up points.

Findings

Refined estimates for blow-up sequences near simple points

Inclusion of local Taylor expansion effects of scalar curvature

Global effects and balance formulas significantly influence blow-up analysis

Abstract

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen apply a refined blow-up analysis (what we call `second order blow-up argument' in this article) to obtain highly accurate approximate solutions for the Yamabe equation. As for the conformal scalar curvature equation on S^n with n > 3, we examine the second order blow-up argument and obtain refined estimate for a blow-up sequence near a simple blow-up point. The estimate involves local effect from the Taylor expansion of the scalar curvature function, global effect from other blow-up points, and the balance formula as expressed in the Pohozaev identity in an essential way.