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Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones

We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions of higher genus SL cones in $\C^{3}$. We study in detail the geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, in preparation for their application to our higher dimensional special Legendrian gluing constructions. In particular the symmetries of these cones and their asymptotics near the spherical limit are analysed. All $\sorth{p} \times \sorth{q}$-invariant SL cones arise from a more general construction of independent interest which we call the special Legendrian twisted product construction. Using this twisted product construction and simple variants of it we can construct a constellation of new special Lagrangian and Hamiltonian stationary cones in $\C^{n}$. We prove the following theorems: A. there are infinitely many topological types of special Lagrangian and Hamiltonian stationary cones in $\C^{n}$ for all $n\ge 4$, B. for $n\ge 4$ special Lagrangian and Hamiltonian stationary torus cones in $\C^{n}$ can occur in continuous families of arbitrarily high dimension and C. for $n\ge 6$ there are infinitely many topological types of special Lagrangian and Hamiltonian stationary cones in $\C^{n}$ that can occur in continuous families of arbitrarily high dimension.