Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric \omega with cusp singularities along a divisor D, we show the L^ 2 projective embedding of (X, D) defined by L^ k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Khler metrics with singularities.

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensions were proved. We shall explain the results obtained as well as the techniques involved. The potential applications in symplectic topology and mirror symmetry will also be discussed.

The authors proved in [5] that in a complete, connected, and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. In this paper, we show that this proposition remains true, if the condition is restricted to balls of equal radii.

We discuss the uniruledness of various base loci of linear systems related to the canonical divisor. In particular we prove that the stable base locus of the canonical divisor of a smooth projective variety of general type is covered by rational curves.

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperk¨ahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperk¨ahler manifolds. We give a Pl¨ucker type formula for Lagrangian intersections under this transformation.