zbMATH — the first resource for mathematics

Biharmonic hypersurfaces in Riemannian manifolds. (English) Zbl 1205.53066
The paper presents some interesting results concerning biharmonic hypersurface in a generic Riemannian manifold. A biharmonic map is a map \(\varphi: (M, g) \to (N, h)\) between Riemannian manifolds that is a critical point of the bienergy functional \(\frac{1}{2}{\int}_{\Omega}{|\tau (\varphi)|}^{2}dx\) for every compact subset \(\Omega\) of \(M\), where \(\tau (\varphi) =\text{Trace}_{g}\nabla d\varphi\) is the tension field of \(\varphi\). A submanifold \(M\) of \((N,h)\) is called a biharmonic submanifold if the inclusion map \(i: (M, i^{*}h) \rightarrow (N, h)\) is a biharmonic isometric immersion. B.-Y. Chen [Soochow J. Math. 17, No. 2, 169–188 (1991; Zbl 0749.53037)] introduced the still-open conjecture that any biharmonic submanifold of Euclidean space is minimal. R. Caddeo, S. Montaldo, and C. Oniciuc [Int. J. Math. 12, No. 8, 867–876 (2001; Zbl 1111.53302)] extended Chen’s conjecture to the generalized one as follows: Any biharmonic submanifold in \((N, h)\) is minimal if the Riemannian curvature operator of \((N, h)\) is nonpositive. The author have proved that the generalized Chen’s conjecture holds for totally umbilical hypersurfaces in an Einstein space (Theorem 2.4) which is the main result of the paper. The author have also proved that there is no proper biharmonic Hopf cylinder in the unit 3-sphere \(S^3\) (Corollary 4.2).

53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI Link arXiv