Project/Area Number 
15340023

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Kobe University 
Principal Investigator 
ROSSMAM W.F. Kobe University, Faculty of Science, Assoc. Professor, 理学部, 助教授 (50284485)

CoInvestigator(Kenkyūbuntansha) 
OHNITA Yoshihiro Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (90183764)
GUEST M. Tokyo Metropolitan University, Graduate School of Science, Professor, 東京・大学院理学研究科, 教授 (10295470)
YAMADA Kotaro Kyushu University, Graduate School of Mathematics, Professor, 大学院数理学研究院, 教授 (10221657)
KOKUBU Masatoshi Tokyo Denki University, Department of Natural Science, Assoc., Professor, 工学部, 助教授 (50287439)
INOGUCHI Junichi Utsunomiya University, Faculty of Education, Assoc., Professor, 教育学部, 助教授 (40309886)

Project Period (FY) 
2003 – 2006

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥10,800,000 (Direct Cost: ¥10,800,000)
Fiscal Year 2006: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2005: ¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2004: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2003: ¥2,500,000 (Direct Cost: ¥2,500,000)

Keywords  constant mean curvature / integrable systems / Euclidean space / 3dimensional spherical space / hyperbolic space / 平坦曲面 
Research Abstract 
The following results were obtained: 1) In a joint research project with U. HertrichJeromin, S. Santos and F. Burstall, a suitable definition for discrete constant mean curvature surfaces in 3 dimensional space forms was obtained. Those 3 dimensional space forms consist of Euclidean 3space, spherical 3space and hyperbolic 3space. It was shown that this new definition matches the old definition that is known for the Euclidean case, and this definition is new in the hyperbolic case. Using this definition, discrete Delaunay surfaces were studied, along with their discrete Darboux and Backlund transformations. An important tool in this research was the notion of conserved quantities. The case of smooth surfaces was developed by S. Santos and F. Burstall, while the discrete case was developed by U. HertrichJeromin and myself. 2) In a joint research project with my Ph.D. graduate student N. Sultana, the stability and Morse index of constant mean curvature surfaces of revolution in spherical 3space was studied. Because the axis of such a surface is a closed loop, these surfaces can become close tori, and then they will have finite index. It was shown that all such surfaces are unstable, and that they all have index at least 5, and (depending on the choice of surface) the index can be arbitrarily large. The index is the number of negative eigenvalues of the associated Jacobi operator. 3) In a continuation of a project with M. Kokubu, M. Umehara and K. Yamada, surfaces with constant Gauss curvature 0 in hyperbolic 3space (flat fronts, which can have singularities) were studied. In particular, this year, it was shown that the caustics of such surfaces can have ends with asymptotic behavior described by cycloids.
