For divisors on abelian varieties, Faltings established an optimal bound on the proximity of rational points to the same. We extend this both to the quasiprojective category, where the role of abelian varieties is played by their toroidal extensions, and to holomorphic maps from the line, proving along the way some wholly general dynamic intersection estimates in value distribution theory of independent interest.

The purpose of this note is to complete the classification of metric fibrations in Euclidean space begun in [1]. Building on our techniques there, we show that regardless of dimension, the fibers are always the orbits of a free isometric group action by generalized glide rotations. A key ingredient of the argument is the fact that in the global setting, these fibrations satisfy a strong algebraic rigidity.

We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Ampére equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary.

Motivated by certain problems in general relativity and Riemannian geometry, we study manifolds which are asymptotic to the hyperbolic space in a certain sense. It is shown that an invariant, the so called total mass, can be defined unambiguously. A positive mass theorem is established by using the spinor method.

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M, w). We consider the evolution of Σ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms w^' and w^" that determine different orientations and Σ is symplectic with respect to both w^' and w^", we prove the mean curvature flow of Σ exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kähler manifolds. In particular, the Poincaré-Lelong equation on complete noncompact Kähler manifolds with nonnegative holomorphic bisectional curvature is studied. Several applications are then derived, which include the Steinness of the complete Kähler manifolds with nonnegative curvature and the flatness of a class of complete Kähler manifolds satisfying a curvature pinching condition. Liouville type results for plurisubharmonic functions are also obtained.