If u∈H1(M,N) is a weakly J-holomorphic map from a compact without boundary almost hermitian manifold (M,j,g) into another compact without boundary almost hermitian manifold (N,J,h). Then it is smooth near any point x where Du has vanishing Morrey norm ℳ2,2m-2, with 2m= dim (M). Hence H2m-2measure of the singular set for a stationary J-holomorphic map is zero. Blow-up analysis and the energy quantization theorem are established for stationary J-holomorphic maps. Connections between stationary J-holomorphic maps and stationary harmonic maps are given for either almost Kähler manifolds M and N or symmetric ∇hJ.
Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow