Supported blow-up and prescribed scalar curvature on S n

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 ≥ 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 ≥ 5 with rather natural condition. © 2011 American Mathematical Society.