We prove the mean curvature flow of the graph of a symplectomorphism between Riemann surfaces converges smoothly as time approaches infinity.

We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.

Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the C1 topology.

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

Let M be a closed orientable 3-manifold admitting an H2 × R or gSL2(R) geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to S1, according as the center of 1(M) is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings SF(M) for compact orientable aspherical 3-manifolds. We also prove that when M has an H2×R or gSL2(R) geometry and the base orbifold has underlying manifold the 2sphere with three cone points, the inclusion Isom(M) ! Diff(M) is a homotopy equivalence.