Equivariant gluing constructions of contact stationary Legendrian submanifolds in $${\mathbb {S}^{2n+1}}$$

Butscher, Adrian
May 2009
Calculus of Variations & Partial Differential Equations;May2009, Vol. 35 Issue 1, p57
Academic Journal
A contact-stationary Legendrian submanifold of $${\mathbb {S}^{2n+1}}$$ 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 S0. This paper develops a method for constructing contact-stationary (but not minimal) Legendrian submanifolds of $${\mathbb {S}^{2n+1}}$$ by gluing together configurations of sufficiently many many U(n + 1)-rotated copies of S0. 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 S0 attached to each other by narrow necks and winding a large number of times around $${\mathbb {S}^{2n+1}}$$ before closing up on themselves; and are topologically equivalent to $${\mathbb {S}^1 \times \mathbb {S}^{n-1}}$$ .


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