Singularity theoretical research on Partial Differential Equations and Differential Geometry

偏微分方程与微分几何的奇异性理论研究

• 批准号: 10304003
• 负责人: IZUMIYA Shyuichi
• 金额: $ 12.12万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304003/
• 关键词: The Eikonal equations; Gravitation lens; Minkowski space; ruled surface; four vertices theorem; hyperbolic Gauss map; Lagrangian immersion; Weak solution; ヘルムホルツ方程式; 漸近解; ハミルトン・ヤコビ方程式; 衝撃波; 特異点; ダルブー可展開面; コンピュータグラフィックス; 特性曲線; ローレンツ微分幾何学

In this research project, we established the fundamental results on the propagation of singualnties (or shock waves) for weak solutions of partial differential equations and the construction on new invariants in Differential Geometry as an application of Singularity theory. Those results could not be studied by using the main frame of 20th century's Mathematics. Those results contain the classification of singularities for solutions of the Eikonal equation which appears in the theory of Ocean acoustics, construction of the generalized notion of weak solutions which is a generalization of both of the entropy and the viscosity solutions, the unified treatment on four vertices theorems of curves, the method to construct many mean curvature constant surfaces, construction of the symplectic framework for multiple-plane garvitation lensing and the study of sinuglarities of hyperbolic Gauss maps and lightcone Gauss maps. In the final year, we have given a classification of singular plane curves by symplectic diffeomorphisms and discovered that the difference from the classification by ordinary diffeomorphisms is a symplectic invariant. We have also found relations between special space curves and ruled surfaces. Moreover, we have studied line congruences and have given a characterization of normal line congruences by the notion of Lagrangian congruences.

在本研究项目中，我们建立了关于偏微分方程弱解的奇性（或激波）传播的基本结果，以及作为奇性理论应用的微分几何中新不变量的构造。这些结果不能用20世纪世纪数学的基本框架来研究。这些结果包括海洋声学理论中Eikonal方程解的奇性分类，构造了作为熵解和粘性解的推广的广义弱解概念，统一处理了曲线的四点定理，构造了许多中曲率常曲面的方法，多平面引力透镜辛框架的建立以及双曲高斯映射和光锥高斯映射的正弦性的研究。在最后一年中，我们给出了奇异平面曲线的辛同态分类，并发现与普通的辛同态分类的区别是辛不变量。我们还发现了特殊的空间曲线和直纹曲面之间的关系。此外，我们还研究了线同余，并利用拉格朗日同余的概念给出了正规线同余的一个刻画。

项目成果

S.Izumiya: "Multivalued solutions to the eikonal equation in stratified media"Quarterly of applied mathematics. LIV. 365-390 (2001)

S.Izumiya：“分层介质中的 eikonal 方程的多值解”应用数学季刊。

K.Fukaya: "Floer homology and Gromov-Witten inyariant over integer of general symplectic manifolds -summary-"Advanced Studies in Pure Mathematics. 31. 75-91 (2001)

K.Fukaya：“一般辛流形整数上的弗洛尔同调和格罗莫夫-维滕不变式-摘要-”纯数学高级研究。

S. Izumiya: "Multivalued solutions to the eikonal equation ii stratified media"Quartely of applied mathematics LIV. 365-390 (2001)

S. Izumiya：“eikonal 方程 ii 分层介质的多值解”应用数学季刊 LIV。

S.IZUMIYA: "Generic affine differential geometry of space curves" Proceedings of the Royal Society of Edinburgh. 128A. 301-314 (1998)

S.IZUMIYA：“空间曲线的通用仿射微分几何”爱丁堡皇家学会会议录。

K. Kiyohara: "Two-dimensional geodesic flows having first integrals of higher degree"Math. Ann.. 320. 487-505 (2001)

K. Kiyohara：“具有更高阶第一积分的二维测地流”数学。