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New Examples of Willmore Surfaces in $S^n$

Li, Haizhong; Vrancken, Luc

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

A surface $x:M\to S^n$ is called a Willmore surface if it is a critical surface 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. It is well-known that any minimal surface is a Willmore surface. The first non-minimal 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 non-minimal flat Willmore surfaces in $S^{5}$. Also in the latter case we explicitly include examples.