[Paper Review] Hamiltonian-minimal Lagrangian submanifolds in complex space forms

Ildefonso Castro, Haizhong Li+1 more|arXiv (Cornell University)|Dec 2, 2004

Geometric Analysis and Curvature Flows6 references4 citations

TL;DR

This paper constructs new examples of Hamiltonian-minimal Lagrangian submanifolds in complex space forms—specifically complex Euclidean, projective, and hyperbolic spaces—using Legendrian submanifolds in odd-dimensional spheres and anti-de Sitter spaces. By lifting these via the Hopf fibration and applying a one-parameter family construction, the authors generate explicit H-minimal Lagrangian immersions and embeddings, including cones in complex Euclidean space and minimal Lagrangian submanifolds in complex projective and hyperbolic spaces.

ABSTRACT

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces also appear. Making use of all of them, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space too.

Motivation & Objective

• To construct new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces beyond those with parallel mean curvature.
• To extend the classification of H-minimal Lagrangian submanifolds to higher dimensions and non-simply-connected quotients.
• To generate Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space via geometric lifting.
• To provide explicit one-parameter families of H-minimal Lagrangian embeddings of quotient manifolds such as $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{S}^{n_2}$ and $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{R}\mathbb{H}^{n_2}$.
• To recover and extend known minimal Lagrangian submanifolds as a byproduct of the construction method.

Proposed method

• Utilizes Legendrian immersions in the unit sphere $\mathbb{S}^{2n+1}$ and anti-de Sitter space $\mathbb{H}_1^{2n+1}$ as a foundational geometric tool.
• Applies the Hopf fibration $\Pi: \mathbb{S}^{2n+1} \to \mathbb{C}\mathbb{P}^n$ and $\Pi: \mathbb{H}_1^{2n+1} \to \mathbb{C}\mathbb{H}^n$ to project Legendrian submanifolds into complex space forms.
• Constructs H-minimal Lagrangian immersions via a one-parameter family of maps involving hyperbolic and trigonometric functions, such as $\sinh\rho$, $\cosh\rho$, and complex exponentials.
• Employs a Weierstrass-type representation via solutions to a system of ODEs on $\mathbb{H}_1^3$, specifically $\alpha_j' \bar{\alpha}_j = i \bar{\alpha}_1^{n_1+1} \bar{\alpha}_2^{n_2+1}$, to generate curves used in the construction.
• Derives explicit parametrizations using real integrals for the radial and angular components of the curves, ensuring the resulting maps are well-defined and smooth.
• Applies symmetry conditions (e.g., $\mathbb{Z}_2$-action) to obtain embedded submanifolds from non-simply-connected quotients.

Experimental results

Research questions

• RQ1 Can new families of Hamiltonian-minimal Lagrangian submanifolds be constructed in complex projective and hyperbolic spaces beyond those with parallel mean curvature?
• RQ2 How can Legendrian submanifolds in $\mathbb{S}^{2n+1}$ and $\mathbb{H}_1^{2n+1}$ be used to generate H-minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{P}^n$ and $\mathbb{C}\mathbb{H}^n$?
• RQ3 What explicit one-parameter families of H-minimal Lagrangian embeddings arise from quotient constructions such as $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{S}^{n_2}$ or $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{R}\mathbb{H}^{n_2}$?
• RQ4 Can special Lagrangian cones in $\mathbb{C}^{n+1}$ be constructed from these geometric lifts?
• RQ5 Under what conditions does the construction yield minimal Lagrangian submanifolds rather than just H-minimal ones?

Key findings

• A one-parameter family of H-minimal Lagrangian immersions is constructed in $\mathbb{C}\mathbb{P}^n$ via projection of Legendrian submanifolds from $\mathbb{S}^{2n+1}$, with explicit examples of embeddings of $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{S}^{n_2}$ for $n_1 + n_2 + 1 = n$.
• New H-minimal Lagrangian immersions are constructed in $\mathbb{C}\mathbb{H}^n$ using Legendrian submanifolds in $\mathbb{H}_1^{2n+1}$, including explicit embeddings of $\mathbb{S}^1 \times \mathbb{S}^{n_1} \times \mathbb{R}\mathbb{H}^{n_2}$ modulo $\mathbb{Z}_2$ action.
• Special Lagrangian cones in $\mathbb{C}^{n+1}$ are obtained as links of the constructed Legendrian submanifolds in $\mathbb{S}^{2n+1}$ and $\mathbb{H}_1^{2n+1}$, providing new examples in complex Euclidean space.
• The construction yields new minimal Lagrangian submanifolds in $\mathbb{C}\mathbb{P}^n$ and $\mathbb{C}\mathbb{H}^n$ as special cases when the initial Legendrian submanifolds are C-minimal.
• A one-parameter family of minimal Lagrangian embeddings is obtained in $\mathbb{C}\mathbb{H}^n$ via a solution to a system of ODEs on $\mathbb{H}_1^3$, with explicit parametrization involving integrals of rational functions of $x^2 + \sinh^2\varrho$ and $x^2 + \cosh^2\varrho$.
• The method produces a complete family of H-minimal Lagrangian submanifolds in $\mathbb{C}^{n+1}$, $\mathbb{C}\mathbb{P}^n$, and $\mathbb{C}\mathbb{H}^n$ for arbitrary $n \geq 2$, generalizing known examples in lower dimensions.