We give a local classification of generalized complex structures. About a point, a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic Poisson manifold. We use a Nash-Moser type argument in the style of Conn’s linearization theorem.

We nd a one-parameter family of coordinates f hgh2R which is a deformation of Penner's simplicial coordinate of the decorated Teichmuller space of an ideally triangulated punctured surface (S; T) of negative Euler characteristic. If h > 0, the decorated Teichmuller space in the h coordinate becomes an explicit convex polytope P(T) independent of h, and if h < 0, the decorated Teichmuller space becomes an explicit bounded convex polytope Ph(T) so that Ph(T)  Ph0 (T) if h < h0. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichmuller space is reproduced.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in $T^{2n}$ is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.

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We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge- Amp`ere type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3. We prove a general local existence result for a large class of Monge-Amp`ere equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.