Polyharmonic Almost Complex Structures

We consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold M^2m . Such objects satisfy the elliptic system [ ^m J, J]=0 weakly. We prove a general regularity theorem for semilinear systems in critical dimensions (with critical growth nonlinearities ), which includes the system of polyharmonic almost complex structures in dimension four and six.