[工程科技]On the moduli of constant mean curvature cylinders of finite type in the 3-sphere.pdfVIP

ON THE MODULI OF CONSTANT MEAN CURVATURE CYLINDERS OF FINITE TYPE IN THE 3-SPHERE 8 M. KILIAN AND M. U. SCHMIDT 0 0 2 Abstract. We show that one-sided Alexandrov embedded constant mean curvature cylin- y ders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by a Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are M rotational. 7 1 Introduction ] G Alexandrov [4] proved that there are no compact embedded surfaces with constant mean cur- D. vature (cmc) in Euclidean 3-space R3 other than round spheres. However, while there are no 3 3 h compact minimal surfaces in R , there is an abundance of such in the 3-sphere S . For instance t a 2-spheres in the 3-sphere are minimal precisely when they are great 2-spheres, and Lawson m proved that compact embedded minimal surfaces in S3 exist for every genus [42, 43]. Lawson [ further showed [44] that any embedded minimal torus in S3 is unknotted, and conjectured that 3 2 up to isometry the Clifford torus is the only embedded minimal torus in S . Hsiang and Lawson v [27] proved that the only embedded minimal torus of revolution is the Clifford torus. Further 8 results suggest that an embedded minimal torus indeed has additional symmetrie