Equivariant gluing constructions of contact-stationary Legendrian submanifolds in the (2n+1)-sphere

AbstractA contact-stationary Legendrian submanifold of the (2n+1)-sphere is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact-stationary Legendrian submanifold (actually minimal Legendrian) is the real, equatorial n-sphere S_0. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of the (2n+1)-sphere by gluing together configurations of sufficiently many U(n + 1)-rotated copies of S_0. Two examples of the construction, corresponding to finite cyclic subgroups of U(n+1), are given. 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 the (2n+1)-sphere before closing up on themselves; and are topologically equivalent to the product of a circle with the (n-1)-sphere.