Gap theorems for minimal submanifolds of Euclidean space

Minimal Lagrangian Submanifolds in Complex Euclidean Space

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.

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]...

First Eigenvalue of Submanifolds in Euclidean Space

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.