  In this paper, we proved the mass angular momentum
inequality\cite{D1}\cite{ChrusLiWe}\cite{SZ} for axisymmetric, asymptotically
flat, vacuum constraint data sets with small trace. Given an initial data set
with small trace, we construct a boost evolution spacetime of the Einstein
vacuum equations as \cite{ChOM}. Then a perturbation method is used to solve
the maximal surface equation in the spacetime under certain growing condition
at infinity. When the initial data set is axisymmetric, we get an axisymmetric
maximal graph with the same ADM mass and angular momentum as the given data.
The inequality follows from the known results\cite{D1}\cite{ChrusLiWe}\cite{SZ}
about the maximal graph.

Differential Geometry;General Relativity and Quantum Cosmology