Metrics of constant scalar curvatures conformal to a Riemannian product with a round sphere

TL;DR

This paper investigates the number of constant scalar curvature metrics conformal to a product metric involving a round sphere and another manifold, showing that this number increases at least linearly with the square root of the scalar curvature.

Contribution

It establishes a lower bound on the growth of conformal metrics with constant scalar curvature in a specific product conformal class, using analysis of radial solutions.

Findings

Number of such metrics grows at least linearly with the square root of scalar curvature

Radial solutions of a specific PDE on spheres are key to counting metrics

Provides a link between scalar curvature and solution multiplicity

Abstract

We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $\Delta u -\lambda u + \lambda u^p =0$ on $S^m$, and the number of solutions in terms of $\lambda$.