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Supported Blow-up and prescribed scalar curvature on $S^n$
About this Title
Man Chun Leung, Department of Mathematics, National University of Singapore, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore
Publication: Memoirs of the American Mathematical Society
Publication Year:
2011; Volume 213, Number 1002
ISBNs: 978-0-8218-5337-5 (print); 978-1-4704-0619-6 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00636-2
Published electronically: March 2, 2011
Keywords: Noncompactness,
blow-up,
removable singularity,
scalar curvature.
MSC: Primary 35J60; Secondary 53C21
Table of Contents
Chapters
- 1. Introduction
- 2. The Subcritical Approach
- 3. Simple, Towering, Aggregated and Clustered Blow-ups
- 4. Supported and Collapsed Blow-ups
- 5. Toward Isolated Blow-ups
- 6. Toward Supported Blow-up for $\Delta \,\tilde K (0) > 0$ – Excluding Simple Blow-up
- 7. Excluding Collapsed Isolated Blow-up (${\mbox {Hess}}_o\,\tilde K (0)\,$ Positive Definite)
- 8. Close Up
- 9. Single Simple Blow-up and the Proof of the Main Theorem
Abstract
We expound the notion of supported blow-up and apply it to study the renowned Nirenberg/Kazdan-Warner problem on $S^n$. When $n \ge 5$ and under some mild conditions, we show that blow-up at a point with positive definite Hessian has to be a supported isolated blow-up, which, when combined with a uniform volume bound, is a removable singularity. A new asymmetric condition is introduced to exclude single simple blow-up. These enable us to obtain a general existence theorem for $n \ge 5\,$ with rather natural condition.- Antonio Ambrosetti, Yan Yan Li, and Andrea Malchiodi, A note on the scalar curvature problem in the presence of symmetries, Ricerche Mat. 49 (2000), no. suppl., 169–176. Contributions in honor of the memory of Ennio De Giorgi (Italian). MR 1826222
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