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Studies on minimal surfaces and Simon conjecture

极小曲面与西蒙猜想的研究

基本信息

批准号：
10640063
负责人：
SAKAMOTO Kunio
金额：
$ 1.79万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640063/
关键词：
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

项目摘要

U.Simon conjectured in 1980 that, for a closed connected minimal surface immersed in an n-dimensional unit sphere, if the Gauss curvature is greater than or equal 2/n(n+1) and less than or equal 2/n(n-1), 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 … More 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 4-dimensional compact submanifold is self-dual or anti-self-dual, then it is critical was shown. We also considered 2-dimensionl 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 S-Willmore points of a Willmore surface, especially, represented the Willmore integral of a Willmore sphere immerced in a 6-dimensional 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

U.Simon在1980年指出，对于n维单位球面中的闭连通极小曲面，如果Gauss曲率大于或等于2/n（n+1）且小于或等于2/n（n-1），则它是常曲率曲面。我们部分解决了这个猜想，并发表在Math. Zeit上的文章。本文证明了如果高斯曲率的拉普拉斯算子被某些二次多项式压缩，则猜想成立。在高斯曲率大于等于1/8且小于等于1/6的情况下，研究了高阶正规空间的退化性。同时，我们还得到了一个结果，即如果由直曲线诱导的度规与曲面上诱导的度规之比小于或等于3倍Gauss曲率，则曲面是标准常曲率球面，并得到了一个退化度越大，该比越大的不等式.自1999年以来，我们 ...更多信息研究了常曲率空间中子流形的法曲率张量长度的共形不变量。得到了某些变分问题的第一变分公式，并研究了临界曲面的性质。特别地，证明了四维紧致子流形的法联络是自对偶或反自对偶的，则它是临界的。我们还考虑了二维情况。在曲率张量的长度为非零常数且曲率椭圆为圆的条件下，临界曲面为Willmore曲面，反之亦然。针对这一结果，给出了Willmore曲面上S-Willmore点的对数留数公式，特别是用对数留数和Euler数表示了Willmore球面浸入6维球面的Willmore积分.证明了紧致临界曲面若满足平均曲率法线平行且曲率椭圆为圆的条件，则为常曲率曲面。在证明这一结果时，利用椭圆函数，我们对具有2次或3次特征函数的共圆标量场的曲面进行了分类，并应用于证明。这篇文章被提交给一家期刊。少