Yamabe Problem And Blow-up Analysis Of Conformal Metrics With Prescribed Scalar Curvature

In 1960,Yamabe[1]proposed the famous Yamabe problem,that is,(M,g)is a smooth compact Riemannian manifold without boundary,whether there is a metric conformal to g,such that its scalar curvature is constant.In 1976,Aubin[2]proofed that if the Yamabe invariant satisfies ?(M)<?(Sn),then the Yamabe problem is affirmative.In this paper,we will give a new proof by Blow-up analysis which is proposed by Sacks and Uhlenbeck[3]when they researched the existence of two-dimensional harmonic maps.A related problem is the convergence and Blow-up analysis of a sequence of conformal metrics with prescribed scalar curvature.Aubin[4]proofed that if the sequence occurs Blow-u,it must be at the critical point of f.In this paper,we will give a new proof by Blow-up analysis.