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.

该项目的目的是研究非线性问题，这是从几何变异问题的角度来研究各种几何结构及其模量空间的研究。以下是项目中获得的主要结果的摘要1。 Nishikawa研究了Infinity的Dirichlet问题，用于一般的K-Term Carnot空间之间的谐波图，这些空间是负曲率的均匀空间，作为k-Step nilpotent carnot Lie基团的可解决率扩展。他发现了理想边界上边界值的必要条件，并在给定边界值的情况下证明了解决方案的存在和独特性。2。 Bando研究了在紧凑的Kaehler歧管上稳定的全体形态载体束上的变性现象OD Hermitian-Einstein指标，并证明可以通过将反射式吊杆添加作为边界来压实这些捆绑包的模量空间。3。中川研究了爱因斯坦 - 凯尔勒指标的存在问题，并证明了combrinatial公式，描述了futaki不变性的，并从相应的convex尸体中读取了数据的foric fano orbifolds上的广义杀戮形式。4。纳亚塔尼（Nayatani）以标准方式构建的伪里曼尼亚人指标与伪符号形式结构兼容，在等级的理想边界上，一个局部局部riemannian riemannian对称类型的对称空间。5。 Horihata研究了与谐波图相关的部分微分方程的非线性抛物线系统，并在空间为3维时，基于单调性公式证明了基于单调性公式的弱溶液的部分规律性。6。 Arai证明了对nilpotent Lie组的伪差方程解决方案解决方案的最佳估计，并将其应用于对切向Cauchy-Riemann方程的解决方案的最佳估计值，以强烈的伪convex cr。