Large conformal metrics with prescribed scalar curvature

Abstract

Let (M,g) be an n-dimensional compact Riemannian manifold. Let h be a smooth function on M and assume that it has a critical point ξ∈M such that h(ξ)=0 and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics gλ=uλ4n−2g, with u>0, whose scalar curvature is the prescribed function hλ:=λ2+h, where λ is a small parameter.In the positive case, i.e. when the scalar curvature Rg is strictly positive, we find a family of “bubbling” metrics gλ, where uλ blows up at the point ξ and approaches zero far from ξ as λ goes to zero.In the general case, if in addition we assume that there exists a non-degenerate conformal metric g0=u04n−2g, with u0>0, whose scalar curvature is equal to h, then there exists a bounded family of conformal metrics g0,λ=u0,λ4n−2g, with u0,λ>0, which satisfies u0,λ→u0 uniformly as λ→0. Here, we build a second family of “bubbling” metrics gλ, where uλ blows up at the point ξ and approaches u0 far from ξ as λ goes to zero. In particular, this shows that this problem admits more than one solution.