Rapid polynomial approximation on Stein manifolds

Plateau–Stein Manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f . We show, for instance, that if an X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurfac...

Lefschetz Fibrations on Compact Stein Manifolds

Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n + 2 > 4 admits a Lefschetz fibration over the disk D with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. This generalizes the Stein surface case of n = 1, previo...

Holomorphic Submersions from Stein Manifolds

Holomorphic Submersions from Stein Manifolds

Lagrangian isotopies in Stein manifolds

Studying the space of Lagrangian submanifolds is a fundamental problem in symplectic topology. Lagrangian spheres appear naturally in the Leftschetz pencil picture of symplectic manifolds. In this paper we demonstrate the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some 4-dimensional Stein symplectic manifolds. The most important example is the cotangent bundle of the 2-sp...