We study the Kahler-Ricci flow on a class of projective bundles over a compact Kahler-Einstein manifold. Assuming the initial Kahler metric admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple (Σ,L,[ω0]), under which the P1-fiber collapses along the Kahler-Ricci flow and the projective bundle converges to Σ in the Gromov-Hausdorff sense. Furthermore, the Kahler-Ricci flow must have Type I singularity and is of (Cn ×P1)-type. This generalizes and extends part of Song-Weinkove’s work on Hirzebruch surfaces.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

We study certain natural differential forms [] and their G equivariant extensions on the space of connections. These forms are defined using the family local index theorem. When the base manifold is symplectic, they define a family of symplectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations).

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Let $f : \sum_1 \to \sum_2$ be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of $f$ can be viewed as a Lagrangian submanifold in $ \sum_1 \times \sum_2$. This article discusses a canonical way to deform $f$ along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of $f$ in $ \sum_1 \times \sum_2$. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that $O(3)$ is a deformation retract of the diffeomorphism group of $S^2$ and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group .