Complete minimal submanifolds with nullity in Euclidean spheres

Minimal Lagrangian Submanifolds in Complex Euclidean Space

Complete Embedded Minimal N-submanifolds in C

A classical problem in the theory of minimal submanifolds of Euclidean spaces is to study the existence of a minimal submanifold with a prescribed behavior at infinity, or to determine from the asymptotes the geometry of the whole submanifold. Beyond the intrinsic interest of these questions, they are also of crucial importance when studying the possible singularities of minimal submanifolds in...

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.

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.

Minimal Submanifolds

Contents 1. Introduction 2 Part 1. Classical and almost classical results 2 1.1. The Gauss map 3 1.2. Minimal graphs 3 1.3. The maximum principle 5 2. Monotonicity and the mean value inequality 6 3. Rado's theorem 8 4. The theorems of Bernstein and Bers 9 5. Simons inequality 10 6. Heinz's curvature estimate for graphs 10 7. Embedded minimal disks with area bounds 11 8. Stable minimal surfaces ...