Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann theory

TL;DR

This paper proves that the Yamabe equation on certain Riemannian products has multiple solutions related to the topological complexity of the manifold, with solutions having energies close to the minimum.

Contribution

It establishes a lower bound on the number of solutions to the Yamabe equation using Lusternik-Schnirelmann theory for Riemannian products.

Findings

Number of solutions is at least Cat(M) + 1.

Solutions have energies arbitrarily close to the minimum.

Results apply to small perturbations of the metric.

Abstract

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + \delta h)$ has at least $Cat(M) +1 $ solutions for $\delta$ small enough, where $Cat(M)$ denotes the Lusternik-Schnirelmann-category of $M$. Cat(M) of the solutions obtained have energy arbitrarily close to the minimum.