The existence of Hamiltonian stationary Lagrangian tori in Kähler manifolds of any dimension
type: Item
instance of: scholarly article
title: The existence of Hamiltonian stationary Lagrangian tori in Kähler manifolds of any dimension
zbMATH Open document ID: 1251.53050
author: Yng-Ing Lee
published in: Calculus of Variations and Partial Differential Equations
publication date: 2012-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.3861
review text: 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.
reviewed by: Yuguang Zhang
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 6087759
zbMATH Keywords: Hamiltonian stationary
zbMATH Keywords: Lagrangian tori
zbMATH Keywords: Kähler manifolds
MaRDI profile type: Publication
OpenAlex ID: W2094740617
arXiv ID: 1001.3861
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DOI: 10.1007/S00526-011-0457-0
description: scientific article; zbMATH DE number 6087759