• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

DEFORMATION AND STABILITY OF SURFACES WITH CONSTANT MEAN CURVATURE

Research Project

Project/Area Number 11640200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKYOTO UNIVERSITY OF EDUCATION

Principal Investigator

KOISO Miyuki  FACULTY OF EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (10178189)

Co-Investigator(Kenkyū-buntansha) KOKUBU Masatoshi  TOKYO DENKI UNIVERSITY, DEPARTMENT OF NATURAL SCIENCE, LECTURER, 工学部, 講師 (50287439)
AIYAMA Reiko  UNIVERSITY OF TSUKUBA, INSTITUTE OF MATHEMATICS, LECTURER, 数学系, 講師 (20222466)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
KeywordsCONSTANT MEAN CURVATURE SURFACE / MINIMAL SURFACE / STABILITY OF CONSTANT MEAN CURVATURE SURFACE / STABILITY OF MINIMAL SURFACE / SECOND VARIATION FORMULA / FREE BOUNDARY PROBLEM / DELAUNAY SURFACE / 第2変分公式 / Weierstrass公式 / ガラス写像
Research Abstract

1. Let P be a complete surface in the threee-dimensional euclidean space. Each critical point of the area functional, among all immersed surfaces with boundary in P and with a given volume, is called a stationary immersion for supporting surface P.In the special case where P is a plane, we proved that any stable stationary immersion is an embedding onto a hemisphere.
2. We studied properties and shapes of nodoids, which are the surfaces of Delaunay (surfaces of revolution with constant mean curvature in the threee-dimensional euclidean space) with self-intersections
3. We obtained sufficient conditions for a immersed surface with constant mean curvature (CMC) in the threee-dimensional euclidean space under which it has a CMC-deformation that fixes the boundary. Moreover, we obtained a criterion of the stability for CMC immersions. Both of these are achieved by using the properties of the eigenvalues and the eigenfunctions of the eigenvalue problem associated to the second variation of the area functional. In a certain special case, by combining these results, we obtained a 'visible' way of judging the stability.
4. We proved that the well-known second variation formula of the area function for regular minimal surfaces is valid also for generalized minimal surfaces (minimal surfaces with branch points) for 'good' variations.
5. We derive sufficient conditions for immersed surfaces with constant mean curvature in three-dimensional space forms to be strongly stable.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (37 results)

All Other

All Publications (37 results)

  • [Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary on a plane"Bulletin of Kyoto University of Education, Ser.B. 97. 1-12 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "On the surfaces of Delaunay"Bulletin of Kyoto University of Education, Ser.B. 97. 13-33 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "Deformation and stability of surfaces with Constant mean curvature"Tohoku Mathematical Journal. (to appear). 16

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki koiso: "A note on the stability of minimal surfaces with branch points""Proceedings of the Fifth Pacific Rim Geometry Conference" (July 25-28, 2000, Tohoku University, Japan). (to appear). 8

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surfaces in S^3 and pairs of non-conformal harmonic maps into S^2"Proceedings of the American Mathematical Society. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku mathematical Journal. 52. 95-105 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"Journal of the Mathematical Society of Japan. 52. 877-898 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the de Sitter 3-space"Tsukuba Journal of Mathematics. 24. 189-196 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Mathematical Journal. 52. 415-429 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masatoshi Kokubu: "Hamiltonian systems derived from constant mean curvature surfaces in hyperbolic three-space"Geometriae Dedicada. 77. 253-269 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masatoshi Kokubu: "On a construction of higher codimensional minimal surfaces based on Enneper's surface and the catenoid"京都大学数理解析研究所講究録. 1113. 65-84 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary on a plane"Bulletin of Kyoto University of Education Ser. B. 97. 1-12 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "On the surface of Delaunay"Bulletin of Kyoto University of Education of Ser.B. 97. 13-33 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "Deformation and Stability of surfaces with constant mean curvature"Tohoku Mathematical Journal. to appear. 16

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki koiso: "A note on the Stability of minimal surfaces with branch points""Proceedings of the Fifth Pacific Rims Geometry Conference (July 25-28, 2000, Tohoku University, Japan)", Tohoku Mathematical Publication. to appear. 8

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "A global correspondence between CMC-surface in S^3 and pairs of non-conformal harmonic maps into S^2"Proceedings of the American Mathematical Society. 128. 939-941 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Mathematical Journal. 52. 95-105 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"Journal of the Mathematical Society of Japan. 52. 877-898 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surface with prescribed mean curvature in the de Sitter 3-space"Tsukuba Journal of Mathematics. 24. 189-196 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Reiko Aiyama: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Mathematical Journal. 52. 415-429 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masatoshi Kokubu: "Hamiltonian systems derived from constant mean curvature surfaces in hyperbolic three-space"Geometriae Dedicata. 77. 253-269 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Masatoshi Kokubu: "On a construction of higher codimensional minimal surfaces based on Enneper's surface and the catenoid"RIMS Kokyuroku. 1113. 65-84 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary on a plane"Bulletin of Kyoto University of Education,Ser.B. No.97. 1-12 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Miyuki Koiso: "On the surfaces of Delaunay"Bulletin of Kyoto University of Education,Ser.B. No.97. 13-33 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the 3-sphere"Tohoku Math,J.. 52・1. 95-105 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama: "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: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Math,J.. 52・3. 415-429 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"Journal of the Mathematical Society of Japan. 52・4. 877-898 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Miyuki Koiso: "The uniqueness for stable surfaces of constant mean curvature with free boundary on a plane"Bulletin of Kyoto University of Education,Ser.B. No.97. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Miyuki Koiso: "On the surfaces of Delaunay"Bulletin of Kyoto University of Education,Ser.B. No.97. (2000)

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

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

    • Related Report
      1999 Annual Research Report
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space"Journal of the Mathematical Society of Japan.

    • Related Report
      1999 Annual Research Report
  • [Publications] Reiko Aiyama: "Minimal maps between the hyperbolic disc and generalized Gauss maps of maximal surfaces in the anti-de Sitter 3-space"Tohoku Mathematical Journal.

    • Related Report
      1999 Annual Research Report
  • [Publications] Reiko Aiyama: "Kenmotsu type representation formula for surfaces with prescribed mean curvature in the de Sitter 3-space"Tsukuba Journal of Mathematics.

    • Related Report
      1999 Annual Research Report
  • [Publications] Masatoshi Kokubu: "Hamiltonian systems derived from constant mean curvature surfaces in hyperbolic three-space"Geometriae Dedicata. 77. 253-269 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Masatoshi Kokubu: "On a construction of higher codimensional minimal surfaces based on Enneper's surface and the catenoid"京都大学数理解析研究所構究録. 1113. 65-84 (1999)

    • Related Report
      1999 Annual Research Report

URL: 

Published: 1999-04-01   Modified: 2016-11-11  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi