O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

Abstract

The aim of the paper is to present a classification of nonextendable immersed O(p+1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in the Euclidean space , with p, q > 1 and 2 ≤ r ≤ 1 min{p, q}, by analyzing embeddedness as well as (r – 1)-stability. The case r = 1 and r = 2 were treated in [Alencar, Trans. Amer. Math. Soc. 337: 129–141, 1993] and [Sato, de Souza Neto, Ann. Global Anal. Geom. 29: 221–240, 2006], respectively. Generalizing the seminal work of Bombieri et al. we also present a (r – 1)-stable complete embedded hypersurface of with Hr = 0 and O(p + 1) × O(q + 1)-invariant, where p + q ≥ r + 5, that is not homeomorphic to .