On K-peak solutions for the Yamabe equation on product manifolds

Abstract

Let $(M^n,g)$ and $(X^m,h)$ be closed manifolds $m,n>2$, such that $(X,h)$ has constant positive scalar curvature. We consider the one parameter family of products $(M\times X,g+{\epsilon }^{2}h)$, $\epsilon >0$. We prove that if either the scalar curvature of $g$, $s_g$, is constant or a certain dimensional constant $\beta =0$, there is some function $\Phi :M\to \mathbb{R}$ that depends on $s_g$, the norm of the Ricci curvature of $g$ and the norm of the curvature tensor of $g$; such that if ${\xi }_0$ is a stable, isolated, critical point of $\Phi$, then for each $K\in \mathbb{N}$, there is some ${\epsilon }_0>0$ such that for every $\epsilon \in (0,{\epsilon }_0)$ the subcritical Yamabe equation $-{\epsilon }^2{\Delta }_{g}u+(1+\mathbf{c}{\epsilon }^2{s}_g)u=u^q$ has a positive $K-$peak solution, which concentrates around ${\xi }_0$. Here, $\mathbf{c}=\frac{N-2}{4(N-1)}$, $q=\frac{N+2}{N-2}$ and $N=n+m$. This provides solutions for the Yamabe equation on Riemannian products $(M\times X,g+{\epsilon }^{2}h)$ and covers some remaining cases of previous results which handle the case where $s_g$ has non-degenerate critical points and $\beta \ne 0$.