Hypersurfaces in Esn+1 satisfying ΔH→=λH→ with at most two distinct principal curvatures

A. Arvanitoyeorgos and G. Kaimakamis proposed in [1] the conjecture that: any hypersurface satisfying ΔH→=λH→ in pseudo-Euclidean space Esn+1 of index s has constant mean curvature. In this paper, we prove that the conjecture is true when the hypersurfaces have at most two distinct principal curvatures. Then, we estimate that constant mean curvature, and give its explicit expression for some special cases. As a result, for that of Lorentzian type hypersurfaces which are not minimal, we prove that it must be isoparametric and give classification results.