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


Author: Man Chun Leung
Journal: Trans. Amer. Math. Soc. 370 (2018), 1123-1157
MSC (2010): Primary 35J60; Secondary 53C21
DOI: https://doi.org/10.1090/tran/6983
Published electronically: September 21, 2017
Supplement: Supplementary appendix
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Man Chun Leung
Affiliation: Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
Email: matlmc@nus.edu.sg

DOI: https://doi.org/10.1090/tran/6983
Keywords: Scalar curvature equation, blow-up, balance formula
Received by editor(s): October 21, 2012
Received by editor(s) in revised form: May 23, 2016
Published electronically: September 21, 2017
Additional Notes: e-Appendix is available at \url{https://arxiv.org/pdf/1707.02401.pdf} (pp. 44–83) and from \url{https://doi.org/10.1090/tran/6983} (Supplementary appendix).
Article copyright: © Copyright 2017 American Mathematical Society

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