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Research on geometric invariants and moduli spaces

几何不变量和模空间研究

基本信息

批准号：
07454237
负责人：
SAKANE Yusuke
金额：
$ 0.96万
依托单位：
Osaka 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

On research for geometric structures, associated geometric invariants and moduli spaces, we obtained following results.1) On minimal surfaces in 3-dimensional Euclidean spaces, we consider a method to construct n-end catenoids with prescribed flux and give a gneral formula for the construction. Moreover, we studied this general furmula of algeraic forms as maps from certain algeraic varieties to complex projective spaces, and proved an existence theorem for n-end catenoids with prescribed flux.2) We generalized the classical Steiner symmetrization to surfaces with self-intersections and applied the generalized Steiner symmetrization to several isoperimetric problems.3) On actions of complex reductive algebraic groups on Kahler manifolds, the notion of (semi-) stablity are inroduced. Then, as an analogy of geometric invariant theory, the existence of geomeric quotient are provedin the category of Kahler manifolds and the data which defines the notion of (semi-) stablity are parametrized … More by certain equivariant cohomology of a Kahler manifold.4) We considered a semilinear hyperbolic version of Eells-Sampson's equation with the resistance. When the resistance goes to infinity, we show that the solution of the semilinear equation converges to a solution of the original parabolic Eells-Sampson's equation.5) We show that a natural quadratic form can be defined on the space of holomorphic vector fields of a compact complex manifold with a fixed Kahler class. Applying this to the case of the first chern class, the periodicity of extremal Kahler vector fields are proved.6) On L^P-spaces we intriduced the notion of orthogonality and proved an orthogonal decomposition theorem. Further, certain compactness of moduli spaces of such L^P-spaces are proved.7) A new computer agebraic system are developed to compute Groebner bases of polynomials. Applying this system, we proved the existence of new invariant Einstein metric on flag manifolds.8) On invariant theory of the Bergman kernel for strictly pseudoconvex domains in C^n, new results are obtained. Less

在几何结构、相关几何不变量和模空间的研究方面，我们得到了以下结果：1)在三维欧氏空间的极小曲面上，我们考虑了一种构造具有给定通量的n端链状的方法，并给出了构造的一般公式。2)将经典的Steiner对称化推广到具有自交的曲面上，并将推广的Steiner对称化应用于几个等周问题。3)关于复约化代数群在Kahler流形上的作用，引入了(半)稳定性的概念。然后，作为几何不变量理论的类比，在Kahler流形范畴中证明了几何商的存在性，并对定义(半)稳定性概念的数据进行了参数化…进一步证明了Kahler流形的某些等变上同调。4)我们考虑了带阻力的Eells-Sampson方程的半线性双曲形式。当阻力趋于无穷大时，我们证明了半线性方程的解收敛于原始抛物型Eells-Sampson方程的解。5)我们证明了在具有固定Kahler类的紧致复流形的全纯向量场空间上可以定义自然二次型。将其应用于第一类陈氏空间，证明了极值Kahler向量场的周期性。6)在L P-空间上，我们引入了正交性的概念，证明了一个正交性分解定理。7)发展了一种新的计算多项式的Groebner基的计算机代数系统。8)关于C^n中严格伪凸域的Bergman核的不变理论，得到了新的结果。较少