On cohomogeneity one biharmonic hypersurfaces into the Euclidean space. (English) Zbl 1341.53010
Summary: The aim of this paper is to prove that there exists no cohomogeneity one \(G\)-invariant proper biharmonic hypersurface in the Euclidean space \(\mathbb R^n\), where \(G\) denotes a transformation group which acts on \(\mathbb{R}^n\) by isometries, with codimension two principal orbits. This result may be considered in the context of the Chen conjecture, since this family of hypersurfaces includes examples with up to seven distinct principal curvatures. The paper uses the methods of equivariant differential geometry. In particular, the technique of proof provides a unified treatment for all these \(G\)-actions.
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 58E20 | Harmonic maps, etc. |
Keywords:
biharmonic maps; biharmonic immersions; transformation groups; equivariant differential geometry; cohomogeneity one hypersurfacesReferences:
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