Hamiltonian stationary cones with isotropic links

Jingyi Chen and Yu Yuan

Vol. 295 (2018), No. 2, 317–327

DOI: 10.2140/pjm.2018.295.317

Abstract

We show that any closed oriented immersed Hamiltonian stationary isotropic surface $\Sigma$ with genus $g_{\Sigma }$ in $S^5\subset {ℂ}^3$ 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 $L^{n-1}\subset S^{2n-1}\subset {ℂ}^n$ 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 nonorientable links are also provided.

In memory of Professor Wei-Yue Ding

Keywords

Hamiltonian stationary cone, minimal Legendrian link

Mathematical Subject Classification 2010

Primary: 58J05

Milestones

Received: 2 May 2017

Revised: 10 October 2017

Accepted: 9 January 2018

Published: 11 April 2018