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負曲率多様体への調和写像と曲面

Research Project

Project/Area Number 11740034
Research Category

Grant-in-Aid for Encouragement of Young Scientists (A)

Allocation TypeSingle-year Grants
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

芥川 玲子 (相山 玲子)  筑波大学, 数学系, 講師 (20222466)

Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2000: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywords平均曲率一定曲面 / 双曲空間 / Gauss写像 / 調和写像 / Dirichlet問題 / 部分多様体 / 曲面 / 定曲率空間形 / 表現公式 / 非完備負曲率多様体
Research Abstract

H^3を定曲率-1の3次元双曲空間形とする。H^3を上半空間モデルで考えると、そこへの曲面Mのはめ込みに対して、Gauss写像Gを(Euclid空間E^3内の場合と同様に)定義できる。すなわち、GはMから単位2次元球面S^2への写像であり、normal Gauss写像と呼ばれる。E^3内の曲面と同様に、H^3内の曲面は平均曲率Hとこのnormal Gauss写像によって局所的には(Kenmotsu型表現公式に則って)決まってしまう。H^3内の平均曲率H一定(CMC-H)曲面について考えるとき、そのnormal Gauss写像Gは、E^3内のCMC曲面のGauss写像のようなS^2の標準計量に対する調和写像にはなっていない。しかし、S^2のある特殊な計量h_Hについては調和写像となっている。特に|H|<1の場合は、h_Hはある円周状に特異点を持っており、その内部ではh_Hは負曲率である。
この計量h_Hを自然に拡張して、n次元単位開球D^n上に考えておく。すなわち、(D^n,g)は負曲率で、完備ではないが境界に近づくにつれて曲率は-∞に発散していて大変興味深い。そこで、m次元双曲空間形H^m(のPoincareモデル)から(D^n,g)への調和写像に対する無限円でのDirichlet問題を考え、解の一意性・存在及びreguralityを調べた。
また、これらの結果を応用して、3次元双曲空間形内の完備で平均曲率一定H(|H|<1)の単連結曲面を、上記のDirichlet問題の解として得られる調和写像がそのnormal Gauss写像となるように構成できることを示した。

Report

(2 results)
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (11 results)

All Other

All Publications (11 results)

  • [Publications] Reiko Aiyama and Kazuo Akutagawa: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math.J.. 52. 95-105 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama and Kazuo Akutagawa: "Kenmotsu type representation formula for spacelike surfaces with prescribed mean curvature in the de Sitter 3-space"Tsukuba J.Math.. 24. 189-196 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama,Kazuo Akutagawa and Tom Y.H.Wan: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Math.J.. 52. 425-429 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama and Kazuo Akutagawa: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"J.Math.Soc.Japan. 52. 877-898 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama and Kazuo Akutagawa: "The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space"Calc.Var.Partial Differential Equations. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] R.Aiyama and K.Akutagawa: "Representation formulas for surtaces in EH^3(-C^2) and harmonic maps arising from CMC surfaces"'Harmonic morphisms, Harmonic maps and Related Topics'(C.K.Anand et al.eds.) Research Notes in Mathematics 413. (chapman & Hall/CRG). 275-286 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Aiyama,K.Akutagawa,R.Miyaoka and M.Umehara: "A global correspondence between CMC-surfaces in $^3 and pairs of non-conformal harmonic maps into $^2"Proceedings of the American Mathematical Society. 128(3). 939-941 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Aiyama and K.Akutagawa: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Mathematical Journal 発売予定. 52(1). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Aiyama and K.Akutagawa: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"Journal of the Mathematical Society of Japan. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Aiyama,K.Akutagawa & T.Y.H.Wan: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Mathematical Journal. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Aiyama and K.Akutagawa: "Kenmotsu type representationformula for surfaces with prescribed mean curvature in the de Sitter 3-space"Tsukuba Journal of Mathematics. (発表予定).

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2021-11-25  

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