New Examples of Willmore Surfaces in S n |
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Authors: | Li Haizhong Vrancken Luc |
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Institution: | (1) Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People's Republic of China;(2) LAMATH, ISTV 2, Campus du Mont Houy, Universite de Valenciennes, France |
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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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Keywords: | Willmore surface minimal surface flat surface |
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