Sequences of surfaces in $4$-manifolds

Abstract

Let $(\Sigma_n)$ be a sequence of surfaces immersed in a $4$-manifold $M$ which converges to a branched surface $\Sigma_0$ .\\ We denote by $k^T_p$ (resp. $k^N_p$) the amount of curvature of the tangent bundles $T\Sigma_n$ (resp. normal bundles $N\Sigma_n$) which concentrates around a branch point $p$ of $\Sigma_0$ when $n$ goes to infinity. Alternatively $k^T\pm k^N$ measures how much the twistor degrees drop when we go from $\Sigma_n$ to $\Sigma_0$. For complex algebraic curves, $k^T+k^N=0$..\\ In some instances - 1) if $\Sigma_0$ is made up of at most $3$ branched disks or 2) if $\Sigma_0$ is area minimizing or 3) if the $\Sigma_n$'s are minimal - we show that $-k^T\geq |k^N|$ and we investigate the equality case.\\ When the second fundamental forms of the $\Sigma_n$'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 $\Sigma_n$'s are minimal, $C$ is a complex curve.