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

We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in $\C^{p+q}$ (equivalently special Legendrian or contact stationary submanifolds in $\Sph^{2(p+q)-1}$) are constructed by combining such objects in $\C^p$ and $\C^q$ using a suitable Legendrian curve in $\Sph^3$. We study the geometry of these “twisting” curves and in particular the closing conditions for them. In combination with Carberry–McIntosh’s continuous families of special Legendrian $2$-tori and the authors’ higher genus special Legendrians, this yields a constellation of new SL and Hamiltonian stationary cones in $\C^n$ that are topological products. In particular, for all $n$ sufficiently large we exhibit infinitely many topological types of SL and Hamiltonian stationary cone in $\C^{n}$, which can occur in continuous families of arbitrarily high dimension.

A special case of the twisted product construction yields all $\sorth{p} \times \sorth{q}$-invariant SL cones in $\C^{p+q}$. These SL cones are higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by Haskins and used in our gluing constructions of higher genus SL cones in $\C^3$. $\sorth{p} \times \sorth{q}$-invariant SL cones play a fundamental role as building blocks in gluing constructions of SL cones in high dimensions. We study some basic geometric features of these cones including their closing and embeddedness properties.

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Published 21 March 2012