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

Juan Miguel Ruiz, Areli Vázquez Juárez

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 $(M\times X, g+ε^2 h)$, $ε>0$. Let $ξ_0\in M$ be a stable critical point of a function $Φ:M\rightarrow \mathbb R$, that depends on the scalar curvature, the dimensions $m,n$, the norm of the Ricci curvature and the norm of the curvature tensor of $g$. We prove that, if either the scalar curvature of $g$ is constant or a certain dimensional constant $β=0$, 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 $-ε^2Δu+(1+{\bf{c}}ε^2 s_g)u=u^q$ on $(M\times X, g+ε^2 h)$ has a $K-$peak positive solution. Where, ${\bf{c}}=\frac{N-2}{4(N-1)}$, $N=n+m$, $q=\frac{N+2}{N-2}$ and $s_g$ the scalar curvature of $(M,g)$. This covers some remaining cases of previous results and gives examples of multiplicity of positive solutions for the Yamabe equation.

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

Abstract

Let and be closed manifolds , such that has constant positive scalar curvature. We consider the one parameter family of products , . Let be a stable critical point of a function , that depends on the scalar curvature, the dimensions , the norm of the Ricci curvature and the norm of the curvature tensor of . We prove that, if either the scalar curvature of is constant or a certain dimensional constant , then for each , there is some such that for every the subcritical Yamabe equation on has a peak positive solution. Where, , , and the scalar curvature of . This covers some remaining cases of previous results and gives examples of multiplicity of positive solutions for the Yamabe equation.
Paper Structure (6 sections, 16 theorems, 379 equations)

This paper contains 6 sections, 16 theorems, 379 equations.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Existence result (proof of Theorem 1.1)
  4. Asymptotic expansion of the energy functional
  5. The Finite dimensional reduction
  6. Some technical estimates

Key Result

Theorem 1.1

Suppose that $\beta=0$ or that $s_g$ is constant. Let $\xi_{0}$ be an isolated local $C^1$ stable critical point of the functional $\Phi(\xi)$. Then, for each positive integer $K$, there exists $\epsilon_0=\epsilon_0(K)>0$ such that for each $\epsilon\in (0,\epsilon_0)$ there are points $\xi_1^{\eps and a solution $u_{\epsilon}$ of problem (Yamabe3), so that

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • proof
  • proof
  • Lemma 5.1
  • ...and 19 more