Abstract
A surface x: M → S n is called a Willmore surface if it is a criticalsurface of the Willmore functional ∫ M (S − 2H 2)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S 1(1) and a particularsmall circle in S 2(1), and therefore is contained in S 5(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S n(1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS 1(1), whereas the other one is contained either in S 2(1) or in S 3(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S 5. Also in the latter casewe explicitly include examples.
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Li, H., Vrancken, L. New Examples of Willmore Surfaces in S n . Annals of Global Analysis and Geometry 23, 205–225 (2003). https://doi.org/10.1023/A:1022825513863
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DOI: https://doi.org/10.1023/A:1022825513863