Polyharmonic hypersurfaces into pseudo-Riemannian space forms

Abstract

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form Nm+1t(c) of dimension m+1 and index t (m≥2 and 1≤t≤m). We shall study hypersurfaces Mmt′ which are polyharmonic of order r (briefly, r-harmonic), where r≥3 and either t′=t or t′=t−1. Let A denote the shape operator of Mmt′. Under the assumptions that Mmt′ is CMC and TrA2 is a constant, we shall obtain the general condition which determines that Mmt′ is r-harmonic.