Asian Journal of Mathematics

Isoparametric hypersurfaces and metrics of constant scalar curvature

Guillermo Henry and Jimmy Petean

Full-text: Open access

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

Subjects
Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
Yamabe equation isoparametric hypersurfaces

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


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