Isoparametric hypersurfaces and metrics of constant scalar curvature

Details Isoparametric hypersurfaces and metrics of constant scalar curvature Isoparametric hypersurfaces and metrics of constant scalar curvature

2011-03-01

arxiv.org/abs/1103.0314v1

Mathematics

Differential Geometry

13 pages

Scientific paper

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.

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