## Asian Journal of Mathematics

### Isoparametric hypersurfaces and metrics of constant scalar curvature

#### 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 $q \lt (m + 2)/ (m - 2)$, 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.

#### Article information

Source
Asian J. Math., Volume 18, Number 1 (2014), 53-68.

Dates
First available in Project Euclid: 27 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1409168512

Mathematical Reviews number (MathSciNet)
MR3215339

Zentralblatt MATH identifier
1292.53041

#### Citation

Henry, Guillermo; Petean, Jimmy. Isoparametric hypersurfaces and metrics of constant scalar curvature. Asian J. Math. 18 (2014), no. 1, 53--68. https://projecteuclid.org/euclid.ajm/1409168512