Example 5.5 [03LC]

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Example 5.5 Define $S$ to be the set of linear special Lagrangian 3-planes $\mathbin{\mathbb{R}}^{3}$ in $\mathbin{\mathbb{C}}^{3}$ containing the real line $\bigl\{(0,0,t):t\in\mathbin{\mathbb{R}}\bigr\}$ . Then $S\cong{\mathcal{S}}^{2}$ . Let $L=\bigl\{(x_{1},x_{2},x_{3})\in\mathbin{\mathbb{C}}^{3}:x_{j}\in\mathbin{\mathbb{R}}\bigr\}$ . Then $L\in S$ . Let $\gamma:\mathbin{\mathbb{R}}\rightarrow S\setminus\{L\}$ be a function which is continuous, but not smooth.

For each $(a,b,c)\in\mathbin{\mathbb{R}}^{3}$ , define $\Pi_{a,b,c}$ to be the affine special Lagrangian 3-plane $\gamma(c)+(a,b,ic)$ . It is not difficult to show that there is a unique, continuous special Lagrangian fibration $f:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$ with $f^{-1}(a,b,c)=\Pi_{a,b,c}$ . However, because $\gamma$ is not smooth, $f$ is not smooth.