New examples of Willmore surfaces in S-n

Abstract

A surface x: M → Sn is called a Willmore surface if it is a criticalsurface of the Willmore functional ∫M (S − 2H2)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 S1(1) and a particularsmall circle in S2(1), and therefore is contained in S5(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in Sn(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 beS1(1), whereas the other one is contained either in S2(1) or in S3(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S5. Also in the latter casewe explicitly include examples.