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The Existence of rG Family and tG Family, and Their Geometric Invariants

Author

Listed:
  • Norio Ejiri

    (Department of Mathematics, Meijo University, Tempaku, Nagoya 468-8502, Japan
    Partly supported by JSPS Grant-in-Aid for Scientific Research (C) 15K04859.)

  • Toshihiro Shoda

    (Faculty of Education, Saga University, 1 Honjo-machi, Saga 840-8502, Japan
    Partly supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03616.)

Abstract

In the 1990s, physicists constructed two one-parameter families of compact oriented embedded minimal surfaces in flat three-tori by using symmetries of space groups, called the rG family and tG family. The present work studies the existence of the two families via the period lattices. Moreover, we will consider two kinds of geometric invariants for the two families, namely, the Morse index and the signature of a minimal surface. We show that Schwarz P surface, D surface, Schoen’s gyroid, and the Lidinoid belong to a family of minimal surfaces with Morse index 1.

Suggested Citation

  • Norio Ejiri & Toshihiro Shoda, 2020. "The Existence of rG Family and tG Family, and Their Geometric Invariants," Mathematics, MDPI, vol. 8(10), pages 1-39, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1693-:d:423051
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