Research on geometric invariants and moduli spaces

在对几何结构、相关几何不变量和模空间的研究中，我们得到了以下结果。1)在三维欧几里得空间的极小曲面上，考虑了一种具有规定通量的n端链面形的构造方法，并给出了构造的一般公式。此外，我们研究了从某些阿尔及利亚变体到复射影空间的映射的阿尔及利亚形式的一般公式，并证明了具有规定通量的n端链面的存在性定理。2)将经典斯坦纳对称推广到具有自交曲面，并将其应用于若干等周问题。3)在复约代数群对Kahler流形的作用上，引入了（半）稳定的概念。然后，作为几何不变理论的类比，证明了几何商在Kahler流形范畴内的存在性，并利用Kahler流形的某些等变上同调参数化了定义（半）稳定概念的数据。4)我们考虑了带有阻力的Eells-Sampson方程的半线性双曲版本。当电阻趋于无穷大时，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)证明了在具有固定Kahler类的紧复流形的全纯向量场空间上可以定义一个自然二次型。将此应用于第一类的情形，证明了极值Kahler向量场的周期性。6)在L^ p空间上引入了正交性的概念，证明了正交分解定理。进一步证明了这种L^ p空间的模空间具有一定的紧性。开发了一种新的计算多项式格罗布纳基的计算机代数系统。应用该系统，证明了标志流形上新的不变爱因斯坦度量的存在性。8)关于C^n上严格伪凸区域的Bergman核的不变量理论，得到了新的结果。少

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