Bubbling analysis at the free boundary for approximate \({\varvec{H}}\)-surfaces

We establish the energy identity and no neck property during the blow-up process for a sequence of approximate H-surfaces \(\{u_n\}_{n \in \mathbb {N}}\) ${\{u_n\}}_{n\in \mathbb{N}}$ from a compact Riemann surface M with smooth boundary \(\partial M\) $\partial M$ into a compact Riemannian manifold (N, h) with free boundaries \(u_n(\partial M) \subset \mathcal {K}\) $u_n(\partial M)\subset \mathcal{K}$ lying on a supporting submanifold \(\mathcal {K} \hookrightarrow N\) $\mathcal{K}\hookrightarrow N$ , with uniformly bounded energy and with tension fields uniformly bounded in \(L^2\) $L^2$ .