Deformations of open 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...

Deformations of representations of fundamental groups of open Kähler manifolds

Given a compact Kähler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of π1(U) with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding moduli space of irreducible representations are quadratic. In the course of our proof we exhibit a diffe...

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...