In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Khler Calabi-Yau surfaces or Hopf surfaces.

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities.We prove that any nonconstant CR morphism between two (2n.1)-dimensional strongly pseudoconvex CR manifolds lying in an n-dimensional Stein variety with isolated singularities are necessarily a CR biholomorphism. As a corollary, we prove that any nonconstant self map of (2n . 1)-dimensional strongly pseudoconvex CR manifold is a CR automorphism. We also prove that a finite ′etale covering map between two resolutions of isolated normal singularities must be an isomorphism.

We study the moduli space M of torsion-free G2-structures on a fixed compact manifold M7, and define its associated universal intermediate Jacobian J . We define the Yukawa coupling and relate it to a natural pseudo-K¨ahler structure on J . We consider natural Chern–Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang–Mills connections), and also deformed Donaldson–Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel–Jacobi maps.

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.

No abstract uploaded!