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Supported Blow-Up and Prescribed Scalar Curvature on $S^n$

Man Chun Leung National University of Singapore, Singapore, Republic of Singapore
Available Formats:
Electronic ISBN: 978-1-4704-0619-6
Product Code: MEMO/213/1002.E
List Price: $73.00 MAA Member Price:$65.70
AMS Member Price: $43.80 Click above image for expanded view Supported Blow-Up and Prescribed Scalar Curvature on$S^n$Man Chun Leung National University of Singapore, Singapore, Republic of Singapore Available Formats:  Electronic ISBN: 978-1-4704-0619-6 Product Code: MEMO/213/1002.E  List Price:$73.00 MAA Member Price: $65.70 AMS Member Price:$43.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 2132011; 99 pp
MSC: Primary 35; Secondary 53;

The author expounds the notion of supported blow-up and applies it to study the renowned Nirenberg/Kazdan-Warner problem on $S^n$. When $n \ge 5$ and under some mild conditions, he shows 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 the author to obtain a general existence theorem for $n \ge 5$ with rather natural condition.

• 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
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Volume: 2132011; 99 pp
MSC: Primary 35; Secondary 53;

The author expounds the notion of supported blow-up and applies it to study the renowned Nirenberg/Kazdan-Warner problem on $S^n$. When $n \ge 5$ and under some mild conditions, he shows 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 the author to obtain a general existence theorem for $n \ge 5$ with rather natural condition.

• 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
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