Construction of compact embedded ${\lambda}$-hypersurfaces via isoparametric hypersurfaces

Abstract

We construct closed embedded ${\lambda}$-hypersurfaces in ${\mathbb{R}^{n+1}}$ of topological type ${S^1\times M}$ for any ${\lambda \ge 0}$, where ${M\subset\mathbb{S}^n}$ is an isoparametric hypersurface with equal principal curvature multiplicities. This extends previous results of Angenent, McGrath, and Riedler (for ${\lambda = 0}$) as well as Cheng-Wei and Ross (for ${\lambda>0}$), and in particular completes the missing case of Riedler's construction.