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 $\xi\in M$ such that $h(\xi)=0$ and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics $g_\lambda=u_\lambda^\frac4{n-2}g$, with $u>0$, whose scalar curvature is the prescribed function $h_\lambda=\lambda^2+h$, where $\lambda$ is a small parameter. In the positive case, i.e. when the scalar curvature $R_g$ is strictly positive, we find a family of bubbling metrics $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches zero far from $\xi$ as $\lambda$ goes to zero. In the general case, if in addition we assume that there exists a non-degenerate conformal metric $g_0=u_0^\frac4{n-2}g$, with $u_0>0$, whose scalar curvature is equal to $h$, then there exists a bounded family of conformal metrics $g_{0,\lambda}=u_{0,\lambda}^\frac4{n-2}g$, with $u_{0,\lambda}>0$, which satisfies $u_{0,\lambda}\to u_0$ uniformly as $\lambda\to 0$. Here, we build a second family of bubbling metrics $g_\lambda$, where $u_\lambda$ blows-up at the point $\xi$ and approaches $u_0$ far from $\xi$ as $\lambda$ goes to zero. In particular, this shows that this problem admits more than one solution.