Studies on minimal surfaces and Simon conjecture
Project/Area Number  10640063 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Saitama University 
Principal Investigator 
SAKAMOTO Kunio Saitama University, Faculty of Science Professor, 理学部, 教授 (70089829)

CoInvestigator(Kenkyūbuntansha) 
NAGASE Masayosi Saitama University, Faculty of Science Professor, 理学部, 教授 (30175509)
OKUMURA Masafumi Saitama University, Faculty of Science Professor, 理学部, 教授 (60016053)

Project Period (FY) 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  Minimal surface / Simon conjecture / Willmore surface / normal curvature tensor / Variational problem / Concircular scalar field / 楕円関数 / 曲率 / ガウス曲率 
Research Abstract 
U.Simon conjectured in 1980 that, for a closed connected minimal surface immersed in an ndimensional unit sphere, if the Gauss curvature is greater than or equal 2/n(n+1) and less than or equal 2/n(n1), then it is a surface of constant curvature. We partially solved this conjecture and published the article from Math. Zeit.. In this paper, we proved that if the Laplacian of the Gauss curvature is pinched by certain quadratic polynomials, then the conjecture is true. Moreover, we studied the degeneracy of the higher order normal space in the case that the Gauss curvature is greater than or equal 1/8 and less than or equal 1/6. Also we obtaied a result that if the ratio of the metric induced by the directrix curve and the induced one on the surface is less than or equal to three times the Gauss curvature, then the surface is a standard constant curvature sphere and an inequality which shows that the greater the degree of the degeneracy becomes the ratio becomes greater. Since 1999, we
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studied conformal invariants concerning the length of the normal curvature tensor for submanifolds immersed in a space of constant curvature. We obtained the first variation formula for some variational problem and studied the properties of the critical surfaces. In particular, the result that if the normal connection of a 4dimensional compact submanifold is selfdual or antiselfdual, then it is critical was shown. We also considered 2dimensionl cases. Under the condition that the length of the normal curvature tensor is a nonzero constant and the curvature ellipse is a circle, critical surfaces are Willmore one and vica versa. Concerning this result, we had a logarithmic residue formula about the SWillmore points of a Willmore surface, especially, represented the Willmore integral of a Willmore sphere immerced in a 6dimensional sphere by the logarithmic residue and the Euler number. Moreover, we proved that if a compact critical surface satisfies the condition that the mean curvature normal is parallel and curvature ellipse is a circle, then it is of constant curvature. In the proof of this result, making use of elliptic functions, we classified surfaces admitting a concircular scalar field with characteristic function of degree 2 or 3 and applied this to the proof. The article is submitted to a journal. Less

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