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 $\Phi:M\rightarrow \mathbb{R}$ 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 $\xi_0$ 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 $-\epsilon^2\Delta_g u+(1+{\bf{c}}\epsilon^2 s_g)u=u^q$ has a positive $K-$peak solution, which concentrates around $\xi_0$. 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 $(M\times X, g+\epsilon^2 h)$ and covers some remaining cases of previous results which handle the case where $s_g$ has non-degenerate critical points and $\beta\neq0$.