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Non-holomorphic, stable minimal surfaces in flat tori

Research Project

Project/Area Number 14540074
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNagoya Institute of Technology

Principal Investigator

EJIRI Norio  Nagoya Institute of Technology, Faculty of Engineering, Professor, 工学研究科, 教授 (80145656)

Co-Investigator(Kenkyū-buntansha) OHYAMA Yosiyuki  Tokyo Woman's Christian University, Department of mathematics, Asocciate Professor, 文理学部, 助教授 (80223981)
SAEKI Akihiro  Nagoya Institute of Technology, Faculty of Engineering, Asocciate Professor, 工学研究科, 助教授 (50270997)
ADACHI Tosiaki  Nagoya Institute of Technology, Faculty of Engineering, Professor, 工学研究科, 教授 (60191855)
Project Period (FY) 2002 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2002: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsharmonic map / periodic minimal surface / holomorphic curve / moduli / schottky problem / stable minimal surface / riemann matrix / hyperelliptic Rimann surface / moduli / Shottky problem / stable minimal surface / オイラー関数
Research Abstract

The Bernstein Problem about minimal graphs in an n-dimensional Euclidean space have been solved. After that, many variations of the Bernstein problem were considered. As an generalization, we know that complete, an orientable, stable minimal surface in a 3-dimensional Euclidean space is only a plane. If we consider a compact, oerientable, minimal surface in a 3-dimensional Riemannian flat torus, then we see that it is totally geodesic, because the deformation in the direction of the unit normal vector field makes the area to be small. Micallef proved that a compact, orientabl, stable minimal surface in an in a 4-dimensional Riemannian flat torus is horomorphic for a suitable orthogonal complex structure of the torus. We expect that compact, We expect that compact, orientable stable minimal surface in a 4-dimensional riemannian flat torus is holomorphic for a suitable orthogonal complex structure of the torus. We expect that compact, orientable, stable minimal surface in lower dimension … More al Riemannian flat tori are horomorphic. Recently, Arezzo and Micallef proved the existence of non-holomorphic, compact, orientable, stable minimal surface of genus g (not less than 7,9,10) in a ((2g-2),(2g-4),(2g-6))-dimensional Riemannian flat torus. They expect the existence of a non-holomorphic, compact, orientable, stable minimal surface of genus g (not less than 4) in a 8-dimensional Riemannian flat torus. On the other hand, investigating the difference between stableness and area area-minimizingness of compact, orientablek minimal surface in Riemannian flat tori, Ejiri have given the following problem: Are compact, orientablestable minimal surfaces of genus g in a 2g-dimensional Riemannian flat torus area-minimizing in the same homotopy class? In this research, we obtain the following fact: Let n be a natural number and M_g(n) the subset of the moduli M_g of Riemann surfaces of genus g(not less than 4) where the element of M_g(n) admits a stable minimal immersion in a 7-demen Rimannian flat torus containing n stable minimal surfaces with different areas in the same homotopy class. Then we obtain the following. Theorem M_g(n) is dense in M_g.. Hence, we know that the Arezzo and Micallef conjecture is true and obtain a counter example for Ejiiri's problem Less

Report

(3 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • Research Products

    (4 results)

All Other

All Publications (4 results)

  • [Publications] Norio Ejiri: "A Differential-Geometric Schottky Problem, and Minimal Surfaces in Tori"Contemporary Mathematics. 308. 101-144 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Ejiri Norio: "A Differential-Geometric Schottky Problem, and minimal surfaces in Tori"Contemporary Mathematics. 308. 101-144

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Norio EJIRI: "Differential-Geometric Schottky Problem, and Minimal Surfaces in Tori"Contemporary Mathematics. 308. 101-144 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Toshiaki ADACHI: "Length spectrum of geodesic spheres and in a non-flat complex space form"Joumal of the Mathematical Society of Japan. 54-2. 629-641 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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