Memoirs of the American Mathematical Society 2011; 99 pp; softcover Volume: 213 ISBN10: 0821853376 ISBN13: 9780821853375 List Price: US$69 Individual Members: US$41.40 Institutional Members: US$55.20 Order Code: MEMO/213/1002
 The author expounds the notion of supported blowup and applies it to study the renowned Nirenberg/KazdanWarner problem on \(S^n\). When \(n \ge 5\) and under some mild conditions, he shows that blowup at a point with positive definite Hessian has to be a supported isolated blowup, which, when combined with a uniform volume bound, is a removable singularity. A new asymmetric condition is introduced to exclude single simple blowup. 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 blowups
 Supported and collapsed blowups
 Toward isolated blowups
 Toward supported blowup for \(\Delta \tilde K(0) > 0\)excluding simple blowup
 Excluding collapsed isolated blowup (\(\mathrm{Hess}_o\tilde K(0)\) positive definite)
 Close up
 Single Simple blowup and the proof of the Main Theorem
 Bibliography
