Submanifolds of some Hartogs domain and the complex Euclidean space

Reducibility of complex submanifolds of the complex euclidean space

Let M be a simply connected complex submanifold of C . We prove that M is irreducible, up a totally geodesic factor, if and only if the normal holonomy group acts irreducibly. This is an extrinsic analogue of the well-known De Rham decomposition theorem for a complex manifold. Our result is not valid in the real context, as it is shown by many counter-examples.

Minimal Lagrangian Submanifolds in Complex Euclidean Space

Reconstructing Submanifolds of Euclidean Space

A generalization of the crust algorithm is presented that will reconstruct a smooth d-dimensional submanifold of R. When the point sample meets satisfy a minimal density requirement this reconstruction is homeomorphic to the original submanifold. In fact the reconstructed manifold is ambiently isotopic to the original via an isotopy that moves points a small distance. Also, bounds are given com...

Lagrangian Submanifolds of Euclidean Space

We give an exposition of the result that there is no closed exact Lagrangian submanifold L of (C, ω0) where ω0 is the standard symplectic structure. We show that the assertion is equivalent to the statement that the perturbed Cauchy-Riemann equation ∂̄J0u = g for maps u from the unit disc D to C which map the boundary circle ∂D to L has no solution for some function g0. To do this, we follow [1]...

Construction of Hamiltonian-minimal Lagrangian Submanifolds in Complex Euclidean Space

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.