MiS Preprint Repository

If $u\in H^1(M,N)$ 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 ${\mathcal{M}}^{2,2m-2}$, with $2m=$dim$(M)$. Hence $H^{2m-2}$ measure 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 ${\mathrm{\nabla }}^{h}J$.