Classification results for biharmonic submanifolds in spheres

Abstract

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersur- faces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo- umbilical 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

Harmonic and Biharmonic Maps at Iaşi

“…The space forms were the first ambient spaces investigated regarding to the existence and classification of proper-biharmonic submanifolds. All the known results obtained for proper-biharmonic submanifolds in Euclidean and hyperbolic spaces were non-existence results (see, for example, [8,22,24,33]). Therefore, the following was conjectured.…”

“…The case of biharmonic hypersurfaces with at most three distinct principal curvatures was approached by first proving that proper-biharmonic hypersurfaces with constant mean curvature in S m+1 , m ≥ 2, have positive constant scalar curvature (see [8]). Then, using certain results on isoparametric hypersurfaces, it was proved that there exist no compact properbiharmonic hypersurfaces of constant mean curvature and with three distinct principal curvatures everywhere in the unit Euclidean sphere (see [9]).…”

“…Taking this further, the type (in the sense of B.-Y. Chen) of compact proper-biharmonic submanifolds of constant mean curvature in S n was studied and it turned out that, depending on the value of their mean curvature, these are of 1-type or of 2-type as submanifolds of R n+1 (see [8,11]). We mention here that spherical submanifolds of low finite type were extensively studied, M. Barros and B.-Y.…”

On the Geometry of Biharmonic Submanifolds in Sasakian Space Forms

“…There are several classification results for the proper-biharmonic submanifolds in Euclidean spheres and non-existence results for such submanifolds in space forms N c , c ≤ 0 ( [4], [5], [7], [8], [9], [10], [13]), while in spaces of non-constant sectional curvature only few results were obtained ( [1], [12], [18], [19], [25], [29]). …”

“…Further, the properbiharmonic curves of S n , n > 3, are those of S 3 (up to a totally geodesic embedding). Concerning the hypersurfaces of S n , it was conjectured in [4] that the only properbiharmonic hypersurfaces are the open parts of S n−1 ( 1…”