Geometriae Dedicata

, Volume 155, Issue 1, pp 105–140 | Cite as

Extrinsic diameter of immersed flat tori in S 3

  • Yoshihisa Kitagawa
  • Masaaki UmeharaEmail author
Original Paper


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.


Rigidity Flat torus Clifford torus Wave front Front 3-sphere Mean curvature Gaussian curvature Extrinsic diameter 

Mathematics Subject Classification (2000)

53C45 53C40 53C42 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUtsunomiya UniversityUtsunomiyaJapan
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

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