Equivariant Gluing Constructions of Contact Stationary Legendrian   Submanifolds of the (2n+1)-Sphere
  A contact stationary Legendrian submanifold of $S^{2n+1}$ is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial $n$-sphere $S_0$. This paper develops a method for constructing contact stationary (but not minimal) Legendrian submanifolds of $S^{2n+1}$ by gluing together configurations of sufficiently many $U(n+1)$-rotated copies of $S_0$ at isolated points of suitably transverse intersection. The resulting submanifolds are very symmetric; are geometrically akin to a `necklace' of copies of $S_0$ attached to each other by narrow necks and winding a large number of times around $S^{2n + 1}$ before closing up on itself; and are topologically equivalent to $S^1 \times S^{n-1}$. Moreover, they represent wholly new examples of contact stationary embedded submanifolds of $S^{2n + 1}$ and thus give rise to wholly new examples of embedded Hamiltonian stationary cones in $C^{n+1}$. 
Adrian Butscher