caa a22 4500 445884703 CHVBK 20180317145555.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s10711-011-9580-5 doi (NATIONALLICENCE)springer-10.1007/s10711-011-9580-5 Extrinsic diameter of immersed flat tori in S 3 [Elektronische Daten] [Yoshihisa Kitagawa, Masaaki Umehara] Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S 3. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S 3. In this paper, we give a new method for characterizing immersed flat tori in S 3 with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H 3 and R 3 regarded as wave fronts has been studied. We also give here a formulation of flat tori in S 3 as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π. Springer Science+Business Media B.V., 2011 Rigidity nationallicence Flat torus nationallicence Clifford torus nationallicence Wave front nationallicence Front nationallicence 3-sphere nationallicence Mean curvature nationallicence Gaussian curvature nationallicence Extrinsic diameter nationallicence Kitagawa Yoshihisa Department of Mathematics, Utsunomiya University, Mine-machi, 321-8505, Utsunomiya, Japan aut Umehara Masaaki Department of Mathematics, Graduate School of Science, Osaka University, 560-0043, Toyonaka, Osaka, Japan aut Geometriae Dedicata Springer Netherlands 155/1(2011-12-01), 105-140 0046-5755 155:1<105 2011 155 10711 https://doi.org/10.1007/s10711-011-9580-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10711-011-9580-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kitagawa Yoshihisa Department of Mathematics, Utsunomiya University, Mine-machi, 321-8505, Utsunomiya, Japan aut NATIONALLICENCE 700 1- Umehara Masaaki Department of Mathematics, Graduate School of Science, Osaka University, 560-0043, Toyonaka, Osaka, Japan aut NATIONALLICENCE 773 0- Geometriae Dedicata Springer Netherlands 155/1(2011-12-01), 105-140 0046-5755 155:1<105 2011 155 10711 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer