Properties of Biharmonic Submanifolds in Spheres

Abstract

Abstract. In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

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

“…From (2) we see that CMC surfaces, i.e. surfaces with constant mean curvature, in space forms are biconservative.…”

“…In general, a submanifold is called biconservative if div S 2 = 0. Moreover, the class of biconservative submanifolds includes that of biharmonic submanifolds, which have been of large interest in the last decade (see, for example, [1,2,3,4,9,19,20]). Biharmonic submanifolds are characterized by the vanishing of the bitension field and they represent a generalization of harmonic (minimal) submanifolds.…”

“…We also would like to point out that submanifolds with vanishing tangent part of the bitension field have been considered by Sasahara in [22] where he studied certain 3-dimensional submanifolds in R 6 . In this paper we consider biconservative surfaces in a 3-dimensional space form N 3 (c) of constant sectional curvature c. In this case (1) becomes (2) 2A(grad f ) + f grad f = 0 .…”

“…where r represents the position vector of a point in R 4 . Then G acts also on S 3 by isometries and it can be identified with the group SO (2). Since the orbits of G are circles of S 3 we deduce that X (u, v) Let consider the following model for the hyperbolic space…”

Biharmonic integral $\mathcal{C}$-parallel submanifolds in 7-dimensional Sasakian space forms

“…When x 0 = 0, the submanifold is called mass-symmetric (see [13]). It was proved in [8,9] that, in general, a proper-biharmonic compact constant mean curvature submanifold M m of S n is either a 1-type submanifold of R n+1 with center of mass of norm equal to 1 √ 2 , or is a mass-symmetric 2-type submanifold of R n+1 . Now, using Theorem 3.5 in [4], where all mass-symmetric 2-type integral surfaces in S 5 (1) were determined, and Proposition 4.1 in [11], the result in [31] can be (partially) reobtained.…”