Construction of special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold

TL;DR

This paper develops a method combining the generalized Legendre transform and moment map techniques to construct special Lagrangian submanifolds in hyperkähler manifolds like Taub-NUT and Atiyah-Hitchin, revealing new invariant submanifolds.

Contribution

It introduces a novel approach to construct special Lagrangian submanifolds with symmetry, expressed through ODEs, and applies it to important hyperkähler manifolds, recovering known results and finding new examples.

Findings

Constructed special Lagrangian submanifolds with cohomogeneity-one symmetry.

Derived conditions for special Lagrangian submanifolds as ODEs.

Numerically obtained solution curves for these ODEs.

Abstract

We construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold by combining the generalized Legendre transform approach and the moment map technique. The generalized Legendre transform approach provides a formulation to construct hyperk\"ahler manifolds and can make their Calabi-Yau structures manifest. In this approach, the K\"ahler $2$-forms and the holomorphic volume forms can be written in terms of holomorphic coordinates, which are convenient to employ the moment map technique. This technique derives the condition that a submanifold in the Calabi-Yau manifold is special Lagrangian. For the Taub-NUT manifold and the Atiyah-Hitchin manifold, by the moment map technique, special Lagrangian submanifolds are obtained as a one-parameter family of the orbits corresponding to Hamiltonian action with respect to their K\"ahler 2-forms. The resultant special Lagrangian submanifolds have cohomogeneity-one symmetry. To demonstrate that our method is useful, we recover the conditions for the special Lagrangian submanifold of the Taub-NUT manifold which is invariant under the tri-holomorphic $U(1)$ symmetry. As new applications of our method, we construct special Lagrangian submanifolds of the Taub-NUT manifold and the Atiyah-Hitchin manifold which are invariant under the action of a Lie subgroup of $SO(3)$. In these constructions, our conditions for being special Lagrangian are expressed by ordinary differential equations (ODEs) with respect to the one-parameters. We numerically give solution curves for the ODEs which specify the special Lagrangian submanifolds for the above cases.