Hamiltonian stationary cones with isotropic links

Abstract

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

References in corpus (1)

• Special Lagrangian cones with higher genus links