Classification results for biharmonic submanifolds in spheres

By A. Balmuş, Stefano Montaldo, C. Oniciuc. 2007

168 citations recorded by the source corpus.

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/51c90acfa45ce87e9eda80f52e49e3e5bf3ac8f3

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Biharmonic ideal hypersurfaces in Euclidean spaces (2013, background): “In particular, there exist many results on the generalized Chen’s conjecture (see, for instance, [1–4,26–30]).”
Biharmonic PNMC submanifolds in spheres (2011, background): “We recall that a pseudo-umbilical submanifold Mm, m 6= 4, of codimension two in Sn is proper biharmonic if and only if it is minimal in Sm+1(1/ √ 2) (see [3]).”
Biharmonic submanifolds of $${\mathbb{C}P^n}$$ (2009, background): “The proper-biharmonic submanifolds of a real space form were extensively studied, see, for example, [ 3 , 4, 5, 6, 8, 11].”
Biharmonic hypersurfaces in Riemannian manifolds (2009, background): “All the known examples of biharmonic submanifolds in spheres lead to the conjecture [4]: any biharmonic submanifold in sphere has constant mean curvature; and any proper biharmonic hypersurface in S is an open part of the hypersphere S( 1 √ 2 ), or the generalized Clifford torus S( 1 √ 2 ) × S( 1 √ ”