An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in \({\mathbb C}^2\) related to Oh's conjecture
type: Item
instance of: scholarly article
title: An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in \({\mathbb C}^2\) related to Oh's conjecture
zbMATH Open document ID: 1030.53078
author: Henri Anciaux
published in: Mathematische Zeitschrift
publication date: 2003-03-27T00:00:00Z
review text: The author computes loops integrals on Hamiltonian stationary Lagrangian tori in \(\mathbb{C}^2\), which are symplectic invariants. Then they show an isoperimetric inequality involving these invariants and the area. Finally, it is proved that the canonical torus \(\mathbb{T}^2_{can} =S^1\times S^1\subset \mathbb{C} \times\mathbb{C}\) has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. A surface of \((\mathbb{C}^2,\omega)\), where \(\omega\) is the standard symplectic form, is said to be Lagrangian if the restriction of \(\omega\) on it vanishes, and Hamiltonian stationary if it is a critical point for the area functional under Hamiltonian deformations. HSLT denotes Hamilton stationary Lagrangian tori of \(\mathbb{C}^2\) studied by \textit{F. Hélein} and \textit{P. Romon} [Commun. Anal. Geom. 10, 79-126 (2002; Zbl 1007.53060); Comment. Math. Helv. 75, 668-680 (2000; Zbl 0973.53025)]. \textit{Y. G. Oh} [Invent. Math. 101, 501-509 (1990; Zbl 0721.53060); Math. Z. 212, 175-192 (1993; Zbl 0791.53050)] studied the second variation of a particular HSLT, the canonical torus \(\mathbb{T}^n_{can}\) in \(\mathbb{C}^n\), showed that it is locally area-minimizing, and conjectured that it is in fact a minimizer in its whole isotopy class (i.e. the set of its images by Hamiltonian deformations). This conjecture is the object of the present paper. At first the author proves the following formula \[ \int_s \psi={1 \over 2\pi} \sum_\gamma {\langle Je_s,\gamma \rangle \over \langle J\beta_0, \gamma\rangle} |h_\gamma |^2, \] where \(\psi\) is any primitive of \(\omega,s\) any element of the first homology group of a HSLT \(X\); \(\langle ., \rangle\) and \(J\) denote respectively the scalar product and the standard complex structure on \(\mathbb{C}\), the vector \(e_s\) represents the loop \(s\), and \(\beta_0\) describes the Lagrangian angle function. This formula implies various inequalities between such integrals and the area of \(X\). Among these inequ…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 1889154
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: critical point
zbMATH Keywords: Hamilton stationary Lagrangian tori
zbMATH Keywords: second variation
zbMATH Keywords: homology group
MaRDI profile type: Publication
Wikidata QID: Q123013650
full work available at URL: s00209-002-0438-2
OpenAlex ID: W1995214600
DOI: 10.1007/S00209-002-0438-2
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description: scientific article; zbMATH DE number 1889154