Multiplicity of constant scalar curvature metrics on warped products

TL;DR

This paper investigates the Yamabe problem on warped product manifolds with positive scalar curvature, demonstrating the existence of multiple solutions using bifurcation and spectral theory techniques.

Contribution

It establishes new multiplicity results for constant scalar curvature metrics on warped products by applying bifurcation theory to the Yamabe equation.

Findings

Proves existence of multiple solutions to the Yamabe problem on warped products.

Uses spectral theory to analyze bifurcation points.

Shows multiplicity depends on geometric properties of the manifold.

Abstract

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume. We study the Yamabe equation on these spaces. We use techniques from bifurcation theory, along with spectral theory for warped products, to prove multiplicity results for the Yamabe problem.