# Embedded constant mean curvature tori in the three-sphere

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 2012, Simon Brendle solved the Lawson conjecture by use of "non-collapsing technique". In 1980s, U. Pinkall and I. Sterling conjectured that embedded tori with CMC in three-sphere are surfaces of revolution. In 2012, Ben Andrews and I 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 or S^1(1/2)\times S^1(\sqrt{3}/2). For other values of the mean curvature, there exists embedded torus which is not S^1(r)\times S^1(\sqrt{1-r^2}). As a Corollary, our Theorem have solved the famous Pinkall-Sterling conjecture.