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Supported Blow-Up and Prescribed Scalar Curvature on \(S^n\)
Man Chun Leung, National University of Singapore, Republic of Singapore
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Memoirs of the American Mathematical Society
2011; 99 pp; softcover
Volume: 213
ISBN-10: 0-8218-5337-6
ISBN-13: 978-0-8218-5337-5
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/213/1002
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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.

Table of Contents

  • Introduction
  • The subcritical approach
  • Simple, towering, aggregated and clustered blow-ups
  • Supported and collapsed blow-ups
  • Toward isolated blow-ups
  • Toward supported blow-up for \(\Delta \tilde K(0) > 0\)--excluding simple blow-up
  • Excluding collapsed isolated blow-up (\(\mathrm{Hess}_o\tilde K(0)\) positive definite)
  • Close up
  • Single Simple blow-up and the proof of the Main Theorem
  • Bibliography
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