ScalingStacks

Example 6.11 [03LV]

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Example 6.11 Define u⁡(x,y)=−y​sinh⁡2​xu(x,y)=-y\sinh 2x and v=y−12​cosh⁡2​xv=y-{\textstyle\frac{1}{2}}\cosh 2x on the half-plane y⩾0y\geqslant 0 in ℝ2\mathbin{\mathbb{R}}^{2}. Then u,vu,v satisfy (32). So equation (31), with the additional condition that y=Im(z1​z2)⩾0y=\mathop{\rm Im}(z_{1}z_{2})\geqslant 0, defines an explicit special Lagrangian 3-fold NN in ℂ3\mathbin{\mathbb{C}}^{3}. It turns out (surprisingly) that NN is nonsingular, and is equivalent to one of the SL 3-folds constructed in [12, Ex. 7.4] by evolving paraboloids in ℂ3\mathbin{\mathbb{C}}^{3}.

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