Development and relations between various geometries and integrable systems

各种几何形状和可积系统之间的发展和关系

基本信息

批准号：
16204007
负责人：
MIYAOKA Reiko
金额：
$ 22.38万
依托单位：
Tohoku University (2006)Kyushu University (2004-2005)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

项目摘要

Miyaoka gives a new proof for the Dorfmeister Neher classification theorem on isoparametric hypersurfaces, and as applications of hypersurface geometry, clarifies the topological structure of the anti-self-dual bundle of complex projective plane and complete austere submanifolds, constructs Ricci flat metrics, special Lagrangian submanifolds. She also gets twister fibrations from the geometry of G2 orbits. Iwasaki connects the algebraic formulation of Painleve IV with the ergodic theory of birational maps of algebraic surfaces via Riemann-Hilbert correspondence, and shows the chaotic behavior of non-linear monodoromy. Kajiwara applies the theoretic formulation of the Painleve systems and constructs the determinant formula of the hypergeometric solutions of q-Painleve, and relates it with the solutions of the associate linear problems. Nakayashiki characterizes the coefficients of the series of sigma function by those of defining functions of the algebraic curves. Nagatomo obtains an es … More sential relation between harmonic maps and the Yang-Mills connections, and generalizes Takahashi's theorem, de Carom-Wallach's theorem, and constructs harmonic maps from quaternion Kaehler manifold to Grassmannian manifolds. Yamada-Umehara-Rossman classify the behavior of the ends of complete flat fronts in the hyperbolic 3-space. Fujioka studies integrability and periodicity of the motion of curves in complex hyperbolics which depend on Burger's equation and have descritization. Ishikawa classifies singularities of inproper affine surfaces and surfaces with constant Gauss curvature, and their dual surfaces. He also clarifies moduli of the singularities, and obtains a relation between plane curves and their Legendle curves. Udagawa classifies compact isotropic submanifolds with parallel mean curvature vector wit the sectional curvature. Tamaru proves a fixed point theorem for cohomogeneity one action corresponding to homogeneous hypersurfaces in symmetric spaces of non-compact type. Matsuura studies a development of plane curves depending on KdV equation w..r.t. discrete time. Ikeda studies equi-energy surfaces of characteristic manifod of Whittaker abel group and full Kostant-Toda lattice via micro-local anaysis. Guest investigates harmonic maps, quantum cohomorogy and mirror symmetry, and writes an introductory book Futaki proves the existence of Sasaki-Einstein metrics on some toric Sasakian manifolds, in particular, the existence of compelete Ricci-flat metric on the canonical bundles of toric Fano manifolds. Less

Miyaoka给出了等参超曲面的Dorfmeister Neher分类定理的新证明，并作为超曲面几何的应用，阐明了复射影平面反自对偶丛和完备简朴子流形的拓扑结构，构造了Ricci平面度量、特殊拉格朗日度量她还从 G2 轨道的几何关系中得到了扭曲纤维振动。 Painleve IV 通过黎曼-希尔伯特对应的代数曲面双有理图的遍历理论，并展示了非线性单性的混沌行为 Kajiwara 应用 Painleve 系统的理论公式并构造了 q- 超几何解的行列式。 Painleve 并将其与相关线性问题的解联系起来，用的系数来表征 sigma 函数级数的系数。 Nagatomo 获得了调和映射与 Yang-Mills 连接之间的本质关系，并推广了 Takahashi 定理、de Carom-Wallach 定理，并构造了从四元数 Kaehler 流形到 Grassmannian 流形的调和映射。 Umehara-Rossman 对双曲线中完全平坦锋面的末端行为进行分类3-空间。藤冈研究了复杂双曲曲线运动的可积性和周期性，石川对不适当仿射曲面和具有恒定高斯曲率的曲面的奇异性进行了分类，他还阐明了的模。奇点，并获得平面曲线与其 Legendle 曲线之间的关系，宇田川将紧致各向同性子流形分类为平行。 Tamaru 证明了与非紧型对称空间中的齐次超曲面对应的不动点定理，该定理依赖于 Ikeda 的 KdV 方程。通过微局域研究 Whittaker abel 群和全 Kostant-Toda 晶格特征流形的等能面Guest 研究了调和映射、量子上同调和镜像对称，并撰写了一本介绍性书籍 Futaki 证明了 Sasaki-Einstein 度量在某些环面 Sasakian 流形上的存在性，特别是在环面正则束上存在完整的 Ricci-Flat 度量。法诺流形较少。