Large conformal metrics with prescribed scalar curvature

TL;DR

This paper constructs conformal metrics with prescribed scalar curvature on compact manifolds, demonstrating bubbling phenomena and multiple solutions under certain flatness and non-degeneracy conditions.

Contribution

It introduces new methods to produce bubbling conformal metrics with prescribed scalar curvature, revealing multiple solutions in the geometric problem.

Findings

Existence of bubbling metrics with scalar curvature approaching zero away from a point.

Construction of a bounded family of conformal metrics converging to a non-degenerate solution.

Identification of multiple solutions under flatness and non-degeneracy assumptions.

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.