Hamiltonian stationary tori in Kähler manifolds (Q453745)

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. This paper proves an existence theorem for Hamiltonian stationary Lagrangian tori in Kähler surfaces. More precisely, if \((M, g, \omega)\) is a Kähler surface, and \(p\in M\) a point satisfying a condition involving holomorphic sectional curvatures at \(p\), then there exist smooth embedded Hamiltonian stationary Lagrangian tori near \(p\). The same result is also obtained by \textit{Y.-I. Lee} [Calc. Var. Partial Differ. Equ. 45, No. 1--2, 231--251 (2012; Zbl 1251.53050)] for any dimensional Kähler manifolds.