Embedded constant mean curvature tori in the three-sphere



This a joint work with Ben Andrews. The minimal surface is the surface with constant mean curvature zero. It was conjectured by H. B. Lawson in 1970s that the only embedded minimal torus in three-sphere is the Clifford torus. In 1980s, U. Pinkall and I. Sterling conjectured that embedded tori with CMC in three-sphere are surfaces of revolution. At March of 2012, Simon Brendle of Stanford University solved the Lawson conjecture. At April of 2012, Ben Andrews and Haizhong Li gave a complete classification of CMC embedded tori in the three-sphere. When the constant mean curvature is equal to zero or ±1/ \sqrt{3}, the only embedded torus is the Clifford torus. For other values of the mean curvature, there exists embedded torus which is not the Clifford torus, Ben Andrews and Haizhong Li gave a complete description of such surfaces. As a Corollary, Ben Andrews and Haizhong Li Theorem have solved the famous Pinkall-Sterling conjecture.

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