In this paper, we prove the mass angular momentum inequality [12, 15, 23] for axisymmetric, asymptotically flat, and vacuum 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 [10]. 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 mass and angular momentum as the given data. The inequality follows from the known results [12, 15, 23] about the maximal graph.
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