Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor

Abstract

We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy. We give the (local) classification of biconservative surfaces in three-dimensional space forms

On biconservative surfaces

“…Theorem 4.7. Let M 2 , g be a surface and consider T be a symmetric tensor field of type (1,1). Then any two of the following relations imply each of the others (1) div T = 0;…”

“…If ϕ : (M m , g) → (N n , h) is a biharmonic map and a Riemannian immersion, then M is called a biharmonic submanifold of N . According to D. Hilbert (see [9]), to a functional E we can associate a symmetric tensor field S of type (1,1), called the stress-energy tensor, which is conservative, i.e., div S = 0, at the critical points of E. In the particular case of the bienergy functional E 2 , G. Y. Jiang (see [13]) defined the stress-bienergy tensor S 2 by…”

Biconservative Submanifolds in $$\mathbb {S}^n\times \mathbb {R}$$ S n × R and $$\mathbb {H}^n\times \mathbb {R}$$ H n × R

“…In this case, the condition of biconservative becomes 2A( grad H) + H grad H = 0, where A is the shape operator and H is the mean curvature function of the hypersurface. The case of surfaces in R 3 was considered by Hasanis-Vlachos [11], and surfaces in S 3 and H 3 was studied by Caddeo-Montaldo-Oniciuc-Piu [2]. In the Euclidean space R 3 , these surfaces are rotational.…”

A classification of biconservative hypersurfaces in a pseudo-Euclidean space

“…On the other hand, in order to understand the geometry of biharmonic submanifolds, geometers have shown attention to study geometrical properties of biconservative submanifolds and contributed accordingly [9,11,18,21,23,24]. It has been observed that some authors have called biconservative hypersurfaces as "H-hypersurfaces" [18,24].…”