New examples of Willmore surfaces in Sn

A surface x : M --> S-n is called a Willmore surface if it is a critical surface of the Willmore functional integral(M)(S - 2H(2))dv, where H is the mean curvature and S is the square of the length of the second fundamental form. It is well known that any minimal surface is a Willmore surface. The first nonminimal example of a flat Willmore surface in higher codimension was obtained by Ejiri. This example which can be viewed as a tensor product immersion of S-1(1) and a particular small circle in S-2(1), and therefore is contained in S-5(1) gives a negative answer to a question by Weiner. In this paper we generalize the above mentioned example by investigating Willmore surfaces in S-n(1) which can be obtained as a tensor product immersion of two curves. We in particular show that in this case too, one of the curves has to be S-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 in terms of elliptic functions, thus constructing infinetely many new nonminimal flat Willmore surfaces in S-5. Also in the latter case we explicitly include examples.