Study of Nonlinear problems in Geometry

几何非线性问题的研究

基本信息

批准号：
07454012
负责人：
NISHIKAWA Seiki
金额：
$ 2.94万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

项目摘要

The purpose of this project is to study Nonlinear Problems, arising in researches of deformation of various geometric structures and their moduli spaces, from the point of view of Geometric Variational Problems. The following is the summary of principal results obtained under the project.1. Nishikawa studied the Dirichlet problem at infinity for harmonic maps between general k-term Carnot spaces, which are homogeneous spaces of negative curvature obtained as solvable extension of k-step nilpotent Carnot Lie groups. He found the necessary conditions for the boundary values on ideal boundaries, and proved the existence and uniqueness of solutions when given boundary values are nondegenerate.2. Bando studied the degeneration phenomena od Hermitian-Einstein metrics on stable holomorphic vector bundles over a compact Kaehler manifold, and proved that the moduli spaces of these bundles can be compactified by adding reflexive sheaves as their boundaries.3. Nakagawa studied the existence problem of Einstein-Kaehler metrics, and proved combrinatiorial formulae describing the Futaki invariants and generalized Killing forms on toric Fano orbifolds terms of data read off from their corresponding convex bodies.4. Nayatani constructed in a standard way pseudo-Riemannian metrics compatible with the pseudo-conformal structures on the ideal boundaries of rank one locally Riemannian symmetric space of noncompact type.5. Horihata studied the nonlinear parabolic system of partial differential equations associated with harmonic map, and proved the partial regularity of weak solutions based on the monotonicity formula in the case when the space is 3 dimensional.6. Arai proved the best possible estimate for solutions of pseudo-differential equations on nilpotent Lie groups, and applied it to obtain the best possible estimate for solutions of the tangential Cauchy-Riemann equation on strongly pseudo-convex CR manifolds.

该项目的目的是从几何变分问题的角度研究各种几何结构及其模空间变形研究中出现的非线性问题。本项目取得的主要成果如下： 1． Nishikawa 研究了一般 k 项卡诺空间之间的调和映射的无穷远狄利克雷问题，这些卡诺空间是作为 k 步幂零卡诺李群的可解扩展而获得的负曲率齐次空间。他找到了理想边界上边界值的必要条件，并证明了给定边界值非简并时解的存在性和唯一性。 2． Bando研究了紧凯勒流形上稳定全纯向量丛的Hermitian-Einstein度量的退化现象，并证明了这些丛的模空间可以通过添加自反滑轮作为边界来压缩。 3． Nakakawa研究了Einstein-Kaehler度量的存在性问题，并证明了描述Futaki不变量和广义Killing形式的组合公式在从相应凸体读取的数据的复曲面Fano轨道项上。 4． Nayatani以标准方式构造了与非紧型局部黎曼对称空间的理想边界上的伪共形结构兼容的伪黎曼度量。 5． Horihata研究了与调和映射相关的非线性抛物型偏微分方程组，并在空间为3维的情况下，基于单调性公式证明了弱解的部分正则性。 6． Arai 证明了幂零李群上伪微分方程解的最佳可能估计，并应用它来获得强伪凸 CR 流形上切向柯西-黎曼方程解的最佳可能估计。