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Extrinsic diameter of immersed flat tori in S 3

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An Erratum to this article was published on 01 September 2013

Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Utsunomiya University, Mine-machi, Utsunomiya, 321-8505, Japan

    Yoshihisa Kitagawa

  2. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Masaaki Umehara

Authors
  1. Yoshihisa Kitagawa
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  2. Masaaki Umehara
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Correspondence to Masaaki Umehara.

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An erratum to this article can be found online at http://dx.doi.org/10.1007/s10711-013-9900-z.

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Kitagawa, Y., Umehara, M. Extrinsic diameter of immersed flat tori in S 3 . Geom Dedicata 155, 105–140 (2011). https://doi.org/10.1007/s10711-011-9580-5

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