Classification of proper biharmonic hypersurfaces in pseudo-Riemannian space forms

Abstract

In this paper, we give some examples of proper biharmonic hypersurfaces in de Sitter space S n+1 q (c) and anti-de Sitter space H n+1 q (c), and prove a classification theorem of nondegenerate proper biharmonic hypersurfaces with diagonalizable shape operator and at most two distinct principal curvatures in pseudo-Riemannian space forms N n+1 q (c).

Biharmonic submanifolds of pseudo-Riemannian manifolds

“…Now we give a complete classification of pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures in a pseudo-Riemannian space form, which improves the classifications given in [17]. (C) with diagonalizable shape operator that has at most two distinct principal curvatures is a part of the following…”

“…As an immediate consequence, we have the following corollary, which was stated in [17] without a proof. Proof.…”

“…He also proved that a pseudo-Riemannian Surface x : M 2 t −→ N 3 1 is proper biharmonic if and only if it is a part of the following: [13] classified proper biharmonic pseudo-Riemannian submanifold x : M m −→ R n s (n ≥ 4, s ≥ 1) with parallel mean curvature vector field H and diagonalizable A H to be marginally trapped. Most recently, Liu-Du [17] classified proper biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures in a pseudo-Riemannian Riemannian space form. Finally, Liu-Du-Zhang in [18] gave a classification of space-like biharmonic submanifolds of a pseudo-Riemannian space form, which are pseudo-umbilical or have parallel mean curvature vector fields.…”

“…Proof. By the classification of Liu-Du [17], we know that a nondegenerate proper biharmonic hypersurface M n t −→ N n+1 s (C) with diagonalizable shape operator that has at most two distinct principal curvatures is a part of one of the following hypersurfaces:…”

Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

“…Sasahara in [11] considered biharmonic hypersurfaces M 2 r of S 3 1 ðcÞ and proved that it must be minimal when r = 0, but may not when r = 1. Investigators studied biharmonic hypersurfaces with at most two distinct principal curvatures in S n+1 s ðcÞ whose shape operator is diagonalizable in [6,8] and showed that such hypersurface M n s−1 is minimal, but the hypersurface M n s may not. Naturally, there is a question as to whether any biharmonic hypersurface M n s−1 in de Sitter space S n+1 s ðcÞ is minimal.…”