Curve-straightening in closed Euclidean submanifolds

Polyharmonic submanifolds in Euclidean spaces

B.Y. Chen introduced biharmonic submanifolds in Euclidean spaces and raised the conjecture ”Any biharmonic submanifold is minimal”. In this article, we show some affirmative partial answers of generalized Chen’s conjecture. Especially, we show that the triharmonic hypersurfaces with constant mean curvature are minimal. M.S.C. 2010: 58E20, 53C43.

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]...

Minimal Lagrangian Submanifolds in Complex Euclidean Space

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.