  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.
