In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, ie the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete Khler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level m structure is also injective.

The aim of this paper is to give a proof of the calculation of the flabby class group of a finite cyclic group due to Endo and Miyata [2]. In the next section I will recall the definition and some basic facts about this group. In the final section I will give some examples to show that the invertibility conditions used by Endo and Miyata cannot be removed. I would like to thank M.c. Kang for useful comments and for showing me his results on some related problems.

In this paper, we construct several kinds of new time-periodic solutions of the vacuum Einsteins field equations whose Riemann curvature tensors vanish, keep finite or take the infinity at some points in these space-times, respectively. The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.

In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces.

Based on the proof of Labastida-Mari {} o-Ooguri-Vafa conjecture\cite {lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N $ invariants. Some symmetry properties of the infinite product will also be discussed.