# Hamiltonian stationary cones with isotropic links

## TL;DR

This paper classifies Hamiltonian stationary isotropic surfaces in the 5-sphere, showing conditions under which they are Legendrian and minimal, and extends results to higher-dimensional isotropic submanifolds with Hamiltonian stationary cones.

## Contribution

It provides a classification of Hamiltonian stationary isotropic surfaces and submanifolds in spheres, establishing when they are Legendrian and minimal based on genus and Betti number.

## Key findings

- Genus zero surfaces are Legendrian and minimal.
- Higher genus surfaces have specific Legendrian point counts.
- Results extend to non-orientable links and higher dimensions.

## 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 $g_{\Sigma}\geq1.$ 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.