2025/05/15 by Balázs Csikós, Csikós, Balázs, Márton Horváth +1
Mathematics · Physics and Astronomy · #53C30 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Fixed Point Theorems Analysis #Primary 53C25 #Secondary 53C40 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2505.10608
openalex publication_date 2025/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify isoparametric functions on Damek-Ricci spaces which can be written in terms of the standard coordinates (v,z,t) on the half-space model as a polynomial function divided by t. Regular level sets of the functions in our classification encompass almost all previously known examples of isoparametric hypersufaces in Damek-Ricci space and also yield new ones. For the new examples, the focal varieties are determined and the mean curvatures of the regular level sets are expressed as a function of the distance from the focal variety. We also study the exceptional case of tubes about \mathbb R\mathbf Hk in \mathbb C\mathbf Hk, which are isoparametric, but cannot be obtained as the level sets of any function in our classification. We show that these tubes are level sets of a polynomial function divided by t2, and that analogous functions on Damek-Ricci spaces can be isoparametric only in the case of \mathbb C\mathbf Hk.