A Characterization of the Ejiri Torus in S^5

We conjecture that a Willmore torus having Willmore functional between 2π2 and 2π2 √3 is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri＇s torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S5 attains the minimum 2π2 √3, which indicates our conjecture holds true for Wilhnore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S5. Moreover, similar to Li and Vrancken, we classify all constrained Wilhnore surfaces of tensor product by reducing them with elastic curves in S3. All constrained Willmore tori obtained this way are also shown to bc unstable when the co-dimension is big enough.