Let $X=\{[z_0,\ldots,z_4]\in\C\P^4\,:\,z_0^5+\ldots+z_4^5=0\}$ (the *Fermat quintic*) with its Calabi–Yau structure (which exists by Yau’s solution of the Calabi conjecture since the first Chern class of $X$ vanishes). Let $\sigma$ be the restriction of complex conjugation on $\C\P^4$ to $X$. Then the fixed point set of $\sigma$, which is the real locus in $X$, is a special Lagrangian 3-fold (if it is non-empty). (There is a subtlety here: $\sigma$ is certainly an anti-holomorphic isometric involution for the induced metric on $X$, but this is *not* the same as the Calabi–Yau metric on $X$. Nevertheless, it is the case that $\sigma$ satisfies the conditions of Proposition \[SLfixedpt.prop\].)

There exists a Calabi–Yau metric on $T^*\mathcal{S}^n$ (the Stenzel metric [@Stenzel]) so that the base $\mathcal{S}^n$ is special Lagrangian: When $n=2$ this is a hyperkähler metric called the Eguchi–Hanson metric [@EguchiHanson].

Constructing calibrated submanifolds
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It is easy to construct complex submanifolds in Kähler manifolds algebraically. Constructing other calibrated submanifolds is much more challenging because one needs to solve a nonlinear PDE, even in Euclidean space. There are approaches in Euclidean space and other simple spaces which have involved reducing the problem to ODEs or other problems which do not require PDE (for example, algebraic methods). 