We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and Bcycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa’s mirror conjecture with bundles in this special case.

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 .

This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains. We prove a “low energy” result in all dimensions, in the sense that if normalized energy in a large ball is small enough, then the normalized energy in any interior ball must also be small.

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on Ham (M, ω). As an application we prove a theorem which can be interpreted as stating that this functional is “virtually” a perfect Morse-Bott functional. This can be applied to study the topology and Hofer geometry of Ham(M, ω). We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz[5].