Hamiltonian stationary cones with isotropic links

We show that any closed oriented immersed Hamiltonian stationary isotropic surface $\mathrm{\Sigma }$ with genus $g_{\mathrm{\Sigma }}$ in $S^5\subset {\mathbb{C}}^3$ is (1) Legendrian and minimal if $g_{\mathrm{\Sigma }}=0$; (2) either Legendrian or with exactly $2g_{\mathrm{\Sigma }}-2$ Legendrian points if $g_{\mathrm{\Sigma }}\ge 1.$ In general, every compact oriented immersed isotropic submanifold $L^{n-1}\subset S^{2n-1}\subset$\mathbb{C}$^{n}$suchthatthecone$C\left( L^{n-1}\right) $ is Hamiltonian stationary must be Legendrian and minimal if its first Betti number is zero. Corresponding results for non-orientable links are also provided.