Isoparametric hypersurfaces and metrics of constant scalar curvature. (English) Zbl 1292.53041

Summary: We show the existence of non-radial solutions of the equation \(\Delta u-\lambda u+\lambda u^q=0\) on the round sphere \(S^m\), for \(q < (m + 2) / (m - 2)\), 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.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
Full Text: DOI arXiv