The existence of Hamiltonian stationary Lagrangian tori in Kähler manifolds of any dimension
A Hamiltonian stationary Lagrangian submanifold \(L\) in a Kähler manifold \((M, g, \omega)\) is a Lagrangian submanifold that is a critical point of the volume functional under Hamiltonian deformations (cf. [\textit{Y.-G. Oh}, Invent. Math. 101, No. 2, 501--519 (1990; Zbl 0721.53060); Math. Z. 212, No. 2, 175--192 (1993; Zbl 0791.53050)]). The Euler-Lagrangian equation for \(L\) is \(d^{*}\alpha_{H}=0\) where \(H\) is the mean curvature vector of \(L\), and \(\alpha_{H}=\omega (H, \cdot)\) on \(L\). Hamiltonian stationary Lagrangian submanifolds are generalizations of minimal Lagrangian submanifolds of Kähler manifolds. The paper under review proves an existence theorem for Hamiltonian stationary Lagrangian tori. More precisely, if \((M, g, \omega)\) is a Kähler manifold of dimension \(n\), and \(p\in M\) is a point satisfying a condition involving a weighted sum of holomorphic sectional curvatures at \(p\), then there exists a smooth family of smooth embedded Hamiltonian stationary Lagrangian tori near \(p\). The same result is also obtained by \textit{A. Butscher} and \textit{J. Corvino} [Calc. Var. Partial Differ. Equ. 45, No. 1--2, 63--100 (2012; Zbl 1251.53048)] for Kähler surfaces.
scientific article; zbMATH DE number 6087759
scientific article