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Classification results for biharmonic submanifolds in spheres. (English) Zbl 1172.58004

A submanifold is biharmonic if the defining isometric immersion is a biharmonic map. Biharmonic maps, as critical points of the bi-energy functional, generalize the notion of harmonic maps, and hence the class of biharmonic submanifolds include minimal submanifolds as a subset. Non-minimal biharmonic submanifolds are called proper biharmonic submanifolds. There may not be any proper biharmonic submanifolds in a Euclidean space as suggested by B.-Y. Chen’s conjecture [Soochow J. Math. 17, No. 2, 169–188 (1991; Zbl 0749.53037)] which states that any biharmonic submanifold in a Euclidean space is minimal. In contrast, proper biharmonic submanifolds of spheres seem to be rich, and they are found to include the generalized Clifford torus ${S}^{p}\left(\frac{1}{\sqrt{2}}\right)×{S}^{q}\left(\frac{1}{\sqrt{2}}\right)↪{S}^{p+q+1},\phantom{\rule{0.277778em}{0ex}}p\ne q$ [G. Jiang, Chin. Ann. Math., Ser. A 7, 130–144 (1986; Zbl 0596.53046)], and the hypersphere ${S}^{n}\left(\frac{1}{\sqrt{2}}\right)↪{S}^{n+1}$ [R. Caddeo, S. Montaldo, and C. Oniciuc, Int. J. Math. 12, No. 8, 867–876 (2001; Zbl 1111.53302)].

The paper under review studies the fundamental problem of classifying biharmonic submanifolds of space forms. Among others, the authors prove that (1) any proper biharmonic hypersurface with at most two distinct principal curvatures in ${S}^{m+1}$ is an open part of ${S}^{m}\left(\frac{1}{\sqrt{2}}\right)$ or of ${S}^{p}\left(\frac{1}{\sqrt{2}}\right)×{S}^{q}\left(\frac{1}{\sqrt{2}}\right),p+q=m,p\ne q$; (2) ${M}^{m}↪{S}^{m+1},m\ge 3,$ is a conformally flat proper biharmonic hypersurface if and only if $M$ is an open part of ${S}^{m}\left(\frac{1}{\sqrt{2}}\right)$ or of ${S}^{1}\left(\frac{1}{\sqrt{2}}\right)×{S}^{m-1}\left(\frac{1}{\sqrt{2}}\right)$; (3) there exists no proper hypersurface with at most two distinct principal curvatures in hyperbolic space ${H}^{m+1}$.

##### MSC:
 58E20 Harmonic maps between infinite-dimensional spaces 53C42 Immersions (differential geometry) 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) 53C43 Differential geometric aspects of harmonic maps
##### References:
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