We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.

We prove the factoriality of a nodal hypersurface in P4 of degree d that has at most 2(d ¡ 1)2=3 singular points, and we prove the factoriality of a double cover of P3 branched over a nodal surface of degree 2r having less than (2r ¡ 1)r singular points.

We prove that Mark Gross’ [8] topological Calabi-Yau compactifications can be made into symplectic compactifications. To prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.

We develop some results from [4] on the positivity of direct image bundles in the particular case of a trivial ¯bration over a one-dimensional base. We also apply the results to study variations of KÄahler metrics.

The weak mean curvature is lower semicontinuous under weak convergence of varifolds, that is, if ¹k ! ¹ weakly as varifolds then k ~H¹ kLp(¹)· lim infk!1 k ~H¹k kLp(¹k). In contrast, if Tk ! T weakly as integral currents, then ¹T may not have a locally bounded ¯rst variation even if k ~H¹Tk kL1(¹k) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semi- continuity of the weak mean curvature is true when T is assumed to be smooth. This was proved in [AmMa03] for p > n = dim T in Rn+1 using results from [Sch04]. Here we prove this in any dimension and codimension down to the desired exponent p = 2. For p = n = 2, this corresponds to the Willmore functional. In a forthcoming joint work [RoSch06], main steps of the pre- sent article are used to prove a modi¯ed conjecture of De Giorgi that the sum of the area and the Willmore functional is the ¡-limit of a di®use Landau-Ginzburg approximation.