Sequences of surfaces in $4$-manifolds

Let $({\mathrm{\Sigma }}_n)$ be a sequence of surfaces immersed in a $4$-manifold $M$ which converges to a branched surface ${\mathrm{\Sigma }}_0$ .\ We denote by $k_p^T$ (resp. $k_p^N$) the amount of curvature of the tangent bundles $T{\mathrm{\Sigma }}_n$ (resp. normal bundles $N{\mathrm{\Sigma }}_n$) which concentrates around a branch point $p$ of ${\mathrm{\Sigma }}_0$ when $n$ goes to infinity. Alternatively $k^T\pm k^N$ measures how much the twistor degrees drop when we go from ${\mathrm{\Sigma }}_n$ to ${\mathrm{\Sigma }}_0$. For complex algebraic curves, $k^T+k^N=0$…\ In some instances - 1) if ${\mathrm{\Sigma }}_0$ is made up of at most $3$ branched disks or 2) if ${\mathrm{\Sigma }}_0$ is area minimizing or 3) if the ${\mathrm{\Sigma }}_n$’s are minimal - we show that $-k^T\ge \mathrm{\mid }{k}^N\mathrm{\mid }$ and we investigate the equality case.\ When the second fundamental forms of the ${\mathrm{\Sigma }}_n$’s have a common $L^2$ bound, we relate $k^T$ and $k^N$ to the bubbling-off of a current $C$ in the Grassmannian $G_2^{+}(M)$. If the ${\mathrm{\Sigma }}_n$’s are minimal, $C$ is a complex curve.