Isoparametric hypersurfaces and metrics of constant scalar curvature

TL;DR

This paper constructs non-radial solutions to a nonlinear PDE on spheres and Riemannian manifolds, revealing multiple metrics of constant scalar curvature associated with isoparametric hypersurfaces.

Contribution

It demonstrates the existence of non-radial solutions linked to isoparametric hypersurfaces, expanding understanding of scalar curvature metrics in conformal geometry.

Findings

Existence of non-radial solutions on spheres for certain nonlinear equations.

Solutions correspond to isoparametric hypersurfaces as level sets.

Results imply multiplicity of constant scalar curvature metrics.

Abstract

We showed the existence of non-radial solutions of the equation $\Delta u -\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $\lambda$. We show that for any isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $\lambda$). We also show similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.