We develop some results from  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.
Gal O L, Sanz F, Speissegger P, et al. Non-interlaced solutions of 2-dimensional systems of linear ordinary differential equations[J]. Proceedings of the American Mathematical Society, 2013, 141(7): 2429-2438.
Dmitry Batenkov · Niv Sarig · Yosef Yomdin. Decoupling of Fourier Reconstruction System for Shifts of Several Signals. 2013.
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.
The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain
stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges
uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue
of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in
C1 to a KÄahler-Einstein metric.
Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial
points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction.
Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry,
by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle
of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological
assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.
We study Dirac structures associated with Manin pairs (d, g) and give a Dirac geometric approach to Hamiltonian spaces with
D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach  in terms of quasi-Poisson structures.
We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of “double” in each context.