Cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere (Q415299)

The authors describe and classify \(SO(p)\times SO(q)\)-invariant cohomogeneity one special Lagrangian submanifolds in the cotangent bundle \(T^* S^n\) of the \(n\)-sphere, when \(p+q=n+1\), by using the moment map as a tool of construction.NEWLINENEWLINEUsing a natural \(SO(n+1)\)-equivariant diffeomorphism, \(T^*S^n\) is identified with the complex quadric \(Q^n=\{z \in \mathbb{C}^{n+1}:\sum_{i=1}^{n+1} z_i^2=1\}\) of \(\mathbb{C}^{n+1}\), endowed with the Stenzel metric, inducing a cohomogeneity-one Calabi-Yau structure with respect to the action of \(SO(n+1)\). This generalizes the cases \(n=2\) of Eguchi and Hanson, and \(n=3\) of Candelas and de la Ossa. Each subgroup \(SO(p)\times SO(q)\) defines an Hamiltonian action on \(T^*S^n\) with moment map \(\mu\). The constructed Lagrangian submanifolds are subsets of level functions \(\mu^{-1}(c)\) that are \(SO(p)\times SO(q)\) invariant, i.e., \(c\) belongs to the center of \(\mathfrak{g}^*\), where \(\mathfrak{g}\) is the Lie algebra of \(SO(p)\times SO(q)\). Several possible cases are considered separately \( 3\leq p\leq q\), \(p=1\) or \(p=2\) and \( q\geq 3\), \(p=q=2\), and \(p=1,~q=2\).NEWLINENEWLINEThe authors define suitable parametrizations of the orbit spaces and the condition for a Lagrangian submanifold \(L\) as an orbit through a curve \(\sigma\) on a level set \(\mu^{-1}(c)\) to be special Lagrangian is reduced to a solution of an explicit first order ODE. Generic special Lagrangian submanifolds are shown to be diffeomorphic to \(\mathbb{R}\times S^{p-1} \times S^{q-1}\), singular orbits and the special Lagrangian submanifolds that are conormal bundles of austere submanifolds of \(S^n\) are described as well.NEWLINENEWLINEFor \(q=2\) and \(p=1\) or \(2\), an explicit construction of an \(S^1\)-family of special Lagrangian foliations is obtained for \(Q^2\) and \(Q^3\), with some singular leaves.NEWLINENEWLINEIn section 4 the authors define a Calabi-Yau structure on the complex cone \(Q^n_0=\{z\in \mathbb{C}^{n+1}: \sum_{i=1}^{n+1}z_i^2=0\}\) that extends the case \(n=3\) of Candelas and la Ossa, and identify the complement \(T^0S^n\) of the zero section in \(T^*S^n\) with this cone (without the origin) using a natural \(SO(n+1)\)-equivariant diffeomorphism, and then construct in a similar way special Lagrangian submanifolds of \(T^0S^n\) that are \(SO(p)\times SO(q)\) invariant. The cone \(Q^n\) is asymptotic to \(Q_0\), and the metric given to \(Q_0^n\) is the limit of the Stenzel metric on \(Q^n\).NEWLINENEWLINEIn Chapter 5 is described the behavior of the ends of these special Lagrangian submanifolds in \(Q^n\), which are asymptotic to special Lagrangian cones in \(Q_0^n\), and how some special Lagrangian submanifolds approach to singular ones under certain limit conditions. In this section the authors also show some figures that indicate how solutions vary under phase change, for some values of \(p,q\).NEWLINENEWLINEThe paper is clear and easy to read.