[Paper Review] On the Lifts of Minimal Lagrangian Submanifolds

Sung Ho Wang|ArXiv.org|Sep 26, 2001

Geometry and complex manifolds3 references3 citations

TL;DR

This paper constructs Calabi-Yau structures on the total space of the canonical line bundle over Kahler-Einstein manifolds, showing that minimal Lagrangian submanifolds lift to special Lagrangian submanifolds in these spaces. It establishes a uniform lower bound for the area of compact minimal Lagrangian submanifolds in complex projective space, and extends the construction to associative, coassociative, and Cayley submanifolds in $G_2$ and $Spin(7)$ manifolds via lifts of superminimal surfaces in self-dual Einstein 4-manifolds.

ABSTRACT

We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated in this setting and hence can be considered as the special Lagrangian lift of $M$. For the integrable $G_2$ and $Spin(7)$ structures on spin bundles and bundles of anti-self-dual 2-forms on self-dual Einstein 4-manifolds constructed by Bryant and Salamon, minimal surfaces with vanishing complex quartic form (super-minimal) admit lifts which are calibrated, i.e., associative, coassociative or Cayley respectively. The lifts in this case can be considered as the tangential lifts or normal lifts of the minimal surface adapted to the quaternionic bundle structure.

Motivation & Objective

• To construct integrable $SU(n+1)$ structures on the total space of the canonical line bundle $\mathbb{L}$ over a Kahler-Einstein manifold $X^n$.
• To show that the canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated, hence a special Lagrangian lift of $M$.
• To establish a uniform lower bound for the area of compact minimal Lagrangian submanifolds in $\mathbb{C}P^n$ with the standard Kahler structure.
• To extend the construction to $G_2$ and $Spin(7)$ structures, showing that superminimal surfaces in self-dual Einstein 4-manifolds lift to associative, coassociative, or Cayley submanifolds.

Proposed method

• Define the tautological $(n,0)$-form $\Upsilon_0$ on the canonical line bundle $\mathbb{L}$ and show $d\Upsilon_0 = -i\gamma \wedge \Upsilon_0$, where $\gamma$ is the trace of the connection 1-forms.
• Construct a Hermitian metric on $\mathbb{L}$ using a radial function $f(r)$ such that the holomorphic volume form $\Upsilon = d\Upsilon_0$ has constant length and is parallel if the metric is Kahler.
• Derive the differential equation $c r + \frac{2}{n} f^{\frac{2}{n}+1} \frac{\partial f}{\partial r} = 0$ to ensure $\Pi$ is closed, leading to the explicit solution $f(r) = \left(-c \frac{n+2}{2} r^2 + c'\right)^{\frac{n}{2n+2}}$.
• Show that the pullback of the $S^1$-bundle $B \to X$ over a minimal Lagrangian $M \subset X$ is flat, and that a well-defined section $s: M \to \phi^*(B)$ exists, which is parallel iff $M$ is minimal.
• Prove that the lift $\tilde{M} \subset B \subset \mathbb{L}$ of a minimal Lagrangian $M$ is special Lagrangian by showing it is calibrated by $\text{Re}(e^{i\theta} \Upsilon)$.
• For superminimal surfaces in self-dual Einstein 4-manifolds, show that the normal and tangent lifts in the $G_2$ and $Spin(7)$ structures are calibrated as associative, coassociative, or Cayley submanifolds.

Experimental results

Research questions

• RQ1 Can the canonical line bundle over a Kahler-Einstein manifold support a Calabi-Yau structure with a parallel holomorphic volume form?
• RQ2 Is the real line bundle over a minimal Lagrangian submanifold in a Kahler-Einstein manifold calibrated, and thus a special Lagrangian lift?
• RQ3 Does the existence of a special Lagrangian lift imply a uniform lower area bound for compact minimal Lagrangian submanifolds in $\mathbb{C}P^n$?
• RQ4 Do superminimal surfaces in self-dual Einstein 4-manifolds admit calibrated lifts in $G_2$ and $Spin(7)$ manifolds constructed via Bryant and Salamon?

Key findings

• The total space of the canonical line bundle $\mathbb{L}$ over a Kahler-Einstein manifold $X^n$ admits a Calabi-Yau structure with a parallel holomorphic volume form $\Upsilon = d\Upsilon_0$, constructed via a radial metric $f(r)$ satisfying $f(r) = \left(-c \frac{n+2}{2} r^2 + c'\right)^{\frac{n}{2n+2}}$.
• The canonical real line bundle $L \subset \mathbb{L}$ over a minimal Lagrangian submanifold $M \subset X$ is calibrated by $\text{Re}(e^{i\theta} \Upsilon)$, making it a special Lagrangian submanifold of $\mathbb{L}$.
• Compact minimal Lagrangian submanifolds in $\mathbb{C}P^n$ admit a uniform lower area bound $\delta(n) > 0$, derived from the isoperimetric inequality applied to their Legendrian lifts in $S^{2n+1}$.
• For a superminimal surface $\Sigma$ in a self-dual Einstein 4-manifold $N$, the normal bundle $\Omega^\perp$ is an associative submanifold in $\bigwedge^2_-N$ under the $G_2$ structure.
• The tangent and normal lifts $\tilde{T}\Sigma$ and $\tilde{N}\Sigma$ in the spin bundle $\mathbb{S} \to N$ are Cayley submanifolds under the $Spin(7)$ structure.
• The construction confirms the existence of many compact calibrated submanifolds in $G_2$ and $Spin(7)$ manifolds, including in $S^4$ and $\mathbb{C}P^2$, via lifts of superminimal surfaces.