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 $(M\times X, g+ε^2 h)$, $ε>0$. We prove that if either the scalar curvature of $g$, $s_g$, is constant or a certain dimensional constant $β=0$, there is some function $Φ:M\rightarrow \mathbb{R}$ 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 $ξ_0$ is a stable, isolated, critical point of $Φ$, then for each $K\in\mathbb{N}$, there is some $ε_0>0$ such that for every $ε\in (0,ε_0)$ the subcritical Yamabe equation $-ε^2Δ_g u+(1+{\bf{c}}ε^2 s_g)u=u^q$ has a positive $K-$peak solution, which concentrates around $ξ_0$. Here, ${\bf{c}}=\frac{N-2}{4(N-1)}$, $q=\frac{N+2}{N-2}$ and $N=n+m$. This provides solutions for the Yamabe equation on Riemannian products $(M\times X, g+ε^2 h)$ and covers some remaining cases of previous results which handle the case where $s_g$ has non-degenerate critical points and $β\neq0$.