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Biharmonic surfaces of \(\mathbb S^{4}\). (English) Zbl 1185.53071

A biharmonic map is a critical point of the bienergy functional defined by the square norm of the tension field, and non-harmonic biharmonic map is called a proper-biharmonic. It follows from Euler-Lagrange equation for the bienergy functional that any harmonic map is also biharmonic, but the converse is not true in general. In this direction, there is a conjecture which says that the only proper-biharmonic hypersurfaces in \(S^n\) are the open parts of hyperspheres \(S^{n-1}(1/\sqrt{2})\) or a generalized Clifford tori \(S^{n_1}(1/\sqrt{2}) \times S^{n_2}(1/\sqrt{2})\), with \(n_1 + n_2 = n-1, n_1 \neq n_2\). Related with this conjecture, one could consider whether there are proper-biharmonic surfaces in \(S^4\) apart from the minimal surfaces of \(S^3(1/\sqrt{2})\).
In this paper, the authors show that the answer for this question is negative in the case of proper-biharmonic surfaces with constant mean curvature in \(S^4\). In other words, they prove that if \(M\) is a proper-biharmonic surface in \(S^4\) with constant mean curvature, then \(M\) is minimal in \(S^3(1/\sqrt{2})\subset S^4\).

MSC:

53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
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