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Biharmonic hypersurfaces in 4-dimensional space forms. (English) Zbl 1210.58013
The authors study proper biharmonic hypersurfaces with at most distinct principal curvatures in space forms. Biharmonic maps between Riemannian manifolds are critical points of the bienergy functional. The authors prove that the only proper biharmonic compact hypersurfaces of $$S^4$$ are the hypersphere $$S^3(\frac{1}{\sqrt{2}})$$ and the torus $$S^1(\frac{1}{\sqrt{2}}) \times S^2(\frac{1}{\sqrt{2}})$$. The strategy to prove the main theorem consists in proving that proper biharmonic hypersurfaces in $$4$$-dimensional space form have constant mean curvature.

##### MSC:
 58E20 Harmonic maps, etc. 53C43 Differential geometric aspects of harmonic maps
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##### References:
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