Geometric structures on manifolds and global variation problems

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On geometric structures on manifolds, global variation problems and moduli spaces, we obtained following research results.1. On harmonic maps, the connectedness and fundamental group of the moduli spaces of all harmonic maps from Riemann sphere to symmetric spaces are investigated and a characterization of harmonic maps into non-compact Lie groups are also studied.2. On complex differential geometry and moduli spaces, actions of reductive algebraic groups on compact Kahler manifolds are studied. In particular, the notion of stability on the action are introduced and existence of quotient space are proved in the category of Kahler manifolds. Further, on main conjectures on value distribution theory of holomorphic curves, a program to investigate this conjecture together with Vojta conjecture in number theory are proposed, and an algebraic geometric proof for second main theorem of Nevanlinna theory are obtained.3. On gage theory including Yang-Mills fields and its application, homology groups of instantons moduli spaces on ALE spaces are studied. And as a generaization of moduli spaces of anti-self-dual connection on ALE spaces, the notion of quiver varieties are introduced and the existence of representations of Kac-Moody algebras of constructible functions on quiver varieties are proved.4. On geomertic structures of dynamical system, the notion of compact Kahler-Liouville manifolds are introduced and it is proved that compact Kahler-Liouville manifolds are toric manifolds. Further, motions of rubber bands are investigated as singular perturbation of semilinear hyperbolic equations and behavior of solutions are studied.5. On affine manifolds and information geometry, it is shown that geometry of Hessian manifolds is closely related to affine differential geometry, information geometry and symplectic geometry.