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New examples of Willmore surfaces in \(S^n\). (English) Zbl 1033.53049

A Willmore surface is a critical surface \(x: M \rightarrow S^{n}\) of the Willmore functional \( \int_{M}(S - 2H^2)\,dv \), where \(H\) is the mean curvature and \(S\) is the square of the length of the second fundamental form. Any minimal surface is a Willmore surface, but the converse is not true, as is seen e.g. by the counter example of Weiner’s open question obtained by N. Ejiri in [Indiana Univ. Math. J. 31, 209–211 (1982; Zbl 0522.53049)]. By generalizing this counter example, the present paper shows that the Willmore surfaces in \(S^{n}(1)\) which can be obtained as a tensor product immersion of two curves, need one of them to be \(S^{1}(1)\) and the other one to be contained either in \(S^{2}(1)\) or in \(S^{3}(1)\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 0522.53049
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