# Solutions of the Yamabe Equation By Lyapunov-Schmidt Reduction

## TL;DR

This paper proves the existence of multiple positive solutions to a Yamabe type equation on closed Riemannian manifolds using Lyapunov-Schmidt reduction and topological methods, with specific results for product manifolds involving spheres.

## Contribution

It introduces a novel application of Lyapunov-Schmidt reduction combined with Morse and Lusternick-Schnirelmann theories to establish multiplicity results for the Yamabe equation on product manifolds.

## Key findings

- Multiple solutions for Yamabe equations on product manifolds.
- At least 2 + 2g solutions for certain surfaces and spheres.
- Results hold for small perturbation parameter  > 0.

## Abstract

Given any closed Riemannian manifold $(M,g)$ we use the Lyapunov-Schmidt finite-dimensional reduction method and the classical Morse and Lusternick-Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on $(M,g)$. If $(N,h)$ is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product $(M\times N , g + \ve^2 h )$, for $\ve >0$ small. For example, if $M$ is a closed Riemann surface of genus ${\bf g}$ and $(N,h) = (S^2 , g_0)$ is the round 2-sphere, we prove that for $\ve >0$ small enough and a generic metric $g$ on $M$, the Yamabe equation on $(M\times S^2 , g + \ve^2 g_0 )$ has at least $2 + 2 {\bf g}$ solutions.