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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.

No abstract uploaded!

No abstract uploaded!

We determine all helix surfaces with parallel mean curvature vector field which are not minimal or pseudo-umbilical in spaces of type $M^{n}(c)\times\mathbb{R}$ , where$M$^{$n$}($c$) is a simply connected$n$-dimensional manifold with constant sectional curvature$c$.