Refined estimates for simple blow-ups of the scalar curvature equation on 𝑆ⁿ

Leung, Man Chun

doi:10.1090/tran/6983

Refined estimates for simple blow-ups of the scalar curvature equation on $S^n$
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Trans. Amer. Math. Soc. 370 (2018), 1123-1157 Request permission

Abstract:

In their work on a sharp compactness theorem for the Yamabe problem, Khuri, Marques and Schoen [J. Differential Geom. 81 (2009), 143–196] 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 $n \ge 4$ , we examine the second order blow - up argument and obtain a refined estimate for a blow - up sequence near a simple blow - up point. The estimate involves the local effect from the Taylor expansion of the scalar curvature function, the global effect from other blow - up points, and the balance formula as expressed in the Pohozaev identity in an essential way.

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