An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in \({\mathbb C}^2\) related to Oh's conjecture
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 inequalities we mention the following: Theorem 1. Given a HSLT \(X\), we denote by \((e_1,e_2)\) the unique basis of its period lattice \(\Delta\) that minimizes the product \(|e_1||e_2|\). Moreover, we assume that \(\langle 2\beta_0,e_2\rangle\) is odd. Then we have the inequality \[ \bigl|(4\pi)^2 \int_{e_1}\psi \int_{e_2}\psi \bigr|\leq\bigl({\mathcal A}(X) \bigr)^2, \] with equality if and only if the torus is monochromatic. The interest of such inequalities comes from, the fact that the periods are symplectic invariants, so they give a way for comparing areas in the same isotopy class. From this the author proves the following main theorem: The torus \(\mathbb{T}^2_{ can}\) is strictly area minimizing among all HSLT of its isotopy class.
scientific article; zbMATH DE number 1889154
scientific article