Special Legendrian submanifolds in toric Sasaki-Einstein manifolds

TL;DR

This paper demonstrates that all toric Sasaki-Einstein manifolds contain special Legendrian submanifolds, constructed via fixed point sets of anti-holomorphic involutions, including examples like $S^2 imes S^3$ and connected sums.

Contribution

It introduces a method to find special Legendrian submanifolds in any toric Sasaki-Einstein manifold using anti-holomorphic involutions.

Findings

Existence of special Legendrian submanifolds in all toric Sasaki-Einstein manifolds.

Construction of a torus $S^1 \times S^1$ in $S^2 \times S^3$.

Extension to special Legendrian submanifolds in connected sums of $S^2 \times S^3$.

Abstract

We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(\tau)\cap S$ of the fixed point set ${\rm fix}(\tau)$ of an anti-holomorphic involution $\tau$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ has a special Legendrian torus $S^{1}\times S^{1}$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$ for each $m\ge 1$.