Closed twisted products and $\sorth{p}\times \sorth{q}$-invariant special Lagrangian cones

Abstract

We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in C (equivalently special Legendrian or contact stationary submanifolds in S2(p+q)−1) are constructed by combining such objects in C and C using a suitable Legendrian curve in S. We study the geometry of these “twisting” curves and in particular the closing conditions for them. In combination with CarberryMcIntosh’s continuous families of special Legendrian 2-tori [3] and the authors’ higher genus special Legendrians [13], this yields a constellation of new special Lagrangian and Hamiltonian stationary cones in C that are topological products. In particular for all n sufficiently large we exhibit infinitely many topological types of SL and Hamiltonian stationary cone in C which can occur in continuous families of arbitrarily high dimension. A special case of the twisted product construction yields all SO(p) × SO(q)-invariant SL cones in C. These SL cones are higher-dimensional analogues of the SO(2)-invariant SL cones constructed previously by Haskins [8,10] and used in our gluing constructions of higher genus SL cones in C [13]. SO(p)× SO(q)-invariant SL cones play a fundamental role as building blocks in gluing constructions of SL cones in high dimensions [14]. We study some basic geometric features of these cones including their closing and embeddedness properties.