Large conformal metrics with prescribed scalar curvature
Given an \(n\)-dimensional compact Riemannian manifold \((M, g),\) \(n\geq 3,\) the prescribed scalar curvature problem (with conformal change of metric) asks to find a metric \(\tilde{g},\) conformal to \(g,\) such that the scalar curvature of \(\tilde{g}\) equals a given function \(h\) on \(M.\) That problem is equivalent to find a solution \(u\) to \[ -\Delta_g u +c(n)R_gu=c(n)hu^p,\quad u>0\;\text{on}\;M, \] where \(\Delta_g=\text{div}_g\nabla_g\) is the Laplace-Beltrami operator, \(c(n)=\frac{n-2}{4(n-1)},\) \(p=\frac{n+2}{n-2}\) and \(R_g\) stands for the scalar curvature associated to the metric \(g.\) Once having the solution \(u,\) it becomes the conformal factor, that is, \(\tilde{g}=u^{\frac{4}{n-2}}g.\) The paper under review deals with the problem of finding conformal metrics \(g_\lambda=u_\lambda^{\frac{4}{n-2}}g,\) whose scalar curvature is the prescribed function \(h_\lambda = \lambda^2 +h,\) with a small parameter \(\lambda,\) and where \(h\) is a smooth function on \(M\) with critical point \(\xi \in M\) such that \(h(\xi) = 0,\) and which satisfies a suitable flatness assumption. In the case when the scalar curvature \(R_g\) is strictly positive, the authors find a family of ``bubbling'' metrics \(g_\lambda\) where \(u_\lambda\) blows up at the point \(\xi\) and approaches zero far from \(\xi\) as \(\lambda\to0.\) In the general case, assuming additionally the existence of a non-degenerate conformal metric \(g_0 = u_0^{\frac{4}{n-2}} g\) whose scalar curvature is equal to \(h,\) it is proved then there exists a bounded family of conformal metrics \(g_{0, \lambda} = u_{0, \lambda}^{\frac{4}{n - 2}} g,\) with \(u_{0, \lambda} > 0,\) for which \(u_{0, \lambda} \to u_0\) uniformly as \(\lambda \to 0.\) In that case the authors build a second family of ``bubbling'' metrics \(g_\lambda,\) with \(u_\lambda\) blowing up at \(\xi\) and approaching \(u_0\) far from \(\xi\) as \(\lambda\to0.\) This shows, in particular, that the problem considered admits more than one solution. The proofs rely on the Lyapunov-Schmidt reduction and fine blow-up analysis.
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