Research on geometric invariants and moduli spaces

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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