Application of integrable systems methods to surfaces with particular variational properties

可积系统方法在具有特定变分特性的表面上的应用

• 批准号: 15340023
• 负责人: ROSSMAM W.F.
• 金额: $ 6.91万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340023/
• 关键词: constant mean curvature; integrable systems; Euclidean space; 3-dimensional spherical space; hyperbolic space; 平坦曲面

The following results were obtained:1) In a joint research project with U. Hertrich-Jeromin, 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 3-space, spherical 3-space and hyperbolic 3-space. 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. Hertrich-Jeromin 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 3-space 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 3-space (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.

1)在与美国Hertrich-Jeromin、S.Santos和F.Burstall的联合研究项目中，得到了三维空间形式中离散常平均曲率曲面的一个合适的定义。这些空间形式包括欧几里得三维空间、球面三维空间和双曲三维空间。结果表明，这种新定义与欧几里得情形的旧定义相匹配，并且在双曲情形下是新的定义。利用这个定义，研究了离散Delaunay曲面及其离散Darboux和Backlund变换。这项研究的一个重要工具是守恒量的概念。光滑曲面的情形是由S.Santos和F.Burstall发展的，而离散情形是由U.Hertrich-Jeromin和我自己发展的。2)在与我的博士研究生N.Sultana的联合研究项目中，研究了球面三维空间中常平均曲率旋转曲面的稳定性和Morse指数。因为这样的曲面的轴线是一个闭合的环，所以这些曲面可以成为闭合的环面，然后它们将具有有限的指数。结果表明，所有这样的曲面都是不稳定的，并且它们的指数都至少为5，并且(取决于曲面的选择)指数可以任意大。3)在与M.Kokubu，M.Umehara和K.Yamada的一个项目的继续中，研究了双曲空间(平坦面，可以有奇点)中具有常高斯曲率0的曲面。特别是，今年的研究表明，这类曲面的焦散线可以有由摆线描述的渐近行为的末端。

项目成果

Unitarization of monodromy representations and constant mean curvature trinoids in 3‐dimensional space forms

• DOI: 10.1112/jlms/jdm005
• 发表时间: 2004-03
• 期刊: Journal of the London Mathematical Society
• 作者: N. Schmitt;M. Kilian;Shimpei Kobayashi;W. Rossman

Constructing Mean Curvature 1 Surfaces in H^3 with Irregular Ends (共著者 梅原雅顕氏, 山田光太郎氏)

构造具有不规则末端的 H^3 中的平均曲率 1 曲面（合著者 Masaaki Umehara、Kotaro Yamada）

• 期刊: Clay Institute Summer School Proceedings (to appear)(未定)
• 作者: M.Umehara;K.Yamada;N.Sultana;N.Sultana;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossmnan;W.Rossman;W.Rossman;W.Rossman;W.Rossman
• 通讯作者: W.Rossman

Period Problems for Mean Curvature 1 Surfaces in $H^3$ (with M. Umehara, K. Yamada)

$H^3$ 中平均曲率 1 曲面的周期问题（与 M. Umehara、K. Yamada 合作）

• 期刊: Advanced Studies in Pure Mathematics, Surveys on Geometry and Integrable Systems (to appear)
• 作者: M.Umehara;K.Yamada;N.Sultana;N.Sultana;W.Rossman;W.Rossman;W.Rossman
• 通讯作者: W.Rossman

Period Problems for Mean Curvature 1 Surfaces in H^3 with application to surfaces of low total curvature (共著者 梅原雅顕氏, 山田光太郎氏)

H^3 中平均曲率 1 曲面的周期问题及其应用于低总曲率曲面（合著者 Masaaki Umehara 和 Kotaro Yamada）

• 期刊: Advanced Studies in Pure Mathematics (to appear)(未定)
• 作者: M.Umehara;K.Yamada;N.Sultana;N.Sultana;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossman;W.Rossmnan;W.Rossman;W.Rossman;W.Rossman
• 通讯作者: W.Rossman

W.Rossman: "Period Problems for Mean Curvature 1 Surfaces in H^3 with application to surfaces of low total curvature(共著者 梅原雅顕氏、山田光太郎氏)"Advanced Studies in Pure Mathematics. (未定)(to appear). (2004)

W.Rossman：“H^3 中平均曲率 1 曲面的周期问题及其应用于低总曲率曲面（合著者 Masaaki Umehara 和 Kotaro Yamada）”纯数学高级研究（待定）（待发表）。 （2004）