The existence of Hamiltonian stationary Lagrangian tori in Kahler manifolds of any dimension

TL;DR

This paper establishes a local criterion in Kähler manifolds that guarantees the existence of Hamiltonian stationary Lagrangian tori near a point, extending previous results to higher dimensions and specific cases like the Clifford torus.

Contribution

It introduces a new local condition in Kähler manifolds ensuring Hamiltonian stationary Lagrangian tori, applicable in any dimension and different from prior conditions.

Findings

Provides a local existence criterion for Hamiltonian stationary Lagrangian tori.

Extends the existence results to higher dimensions and the Clifford torus case.

Differentiates from previous conditions proposed in dimension two.

Abstract

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They can be considered as a generalization of special Lagrangians or Lagrangian and minimal submanifolds. Joyce, Schoen and the author show that given any compact rigid Hamiltonian stationary Lagrangian in $\C^n$, one can always find a family of Hamiltonian stationary Lagrangians of the same type in any compact symplectic manifolds with a compatible metric. The advantage of this result is that it holds in very general classes. But the disadvantage is that we do not know where these examples locate and examples in this family might be far apart. In this paper, we derive a local condition on Kahler manifolds which ensures the existence of one family of Hamiltonian stationary Lagrangian tori near a point with given frame satisfying the criterion. Butscher and Corvino ever proposed a condition in n=2. But our condition appears to be different from theirs. The condition derived in this paper not only works for any dimension, but also for the Clifford torus case which is not covered by their condition.