2012 # 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.

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

confidence: 94%

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

confidence: 94%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%