  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 $C^{n+1}$.

Differential Geometry;Analysis of PDEs