Rigidity of a class of special Lagrangian fibrations singularity. (Q1811385)

The prototype of \textit{SL}-fibration in \(\mathbb{C}^3\) with homogeneous isolated singularity is the example given by \textit{F. R. Harvey} and \textit{H. B. Lawson} in [Acta Math. 148, 47--157 (1982; Zbl 0584.53021)]. In this paper, the author proves the following uniqueness result: Let \(F=(f_1,f_2,f_3):{\mathbb{C}}^3\rightarrow\mathbb{R}^3\) be a homogeneous \textit{SL}-fibration so that its central fiber is a regular cone and \(n_i=\text{ deg} f_i\) be such that \(n_1\leqslant n_2\leqslant n_3\). Then \((n_1,n_2,n_3)=(2,2,3)\) and there is a unitary matrix \(S\) such that if we make the Darboux coordinate change \((p_1,p_2,p_3,q_1,q_2,q_3)^T=S^{-1}(x_1,x_2,x_3,y_1,y_2,y_3)^T\) and let \(w_k=p_k+iq_k\), then \((f_1,f_2,f_3)\) is linearly equivalent to \(\tilde f_1=| w_1| ^2-| w_2| ^2\), \(\tilde f_2=| w_1| ^2-| w_3| ^2\) and \(\tilde f_3=\text{ Im}(w_1w_2w_3)\).