The boundary value problem for triholomorphic maps

We introduce a natural boundary value problem for a triholomorphic map u from a compact almost hyper-Hermitian manifold M with smooth boundary ∂ M into a closed hyperKähler manifold N with free boundary u(∂ M)⊂ Γ lying on some geometrically natural closed supporting submanifold Γ⊂ N , called tri-isotropic submanifold. We establish partial regularity theory and energy quantization result in this boundary setting under some additional assumption on the W2,1 norm of the weakly converging sequences.