Quantitative flatness results for nonlocal minimal surfaces in low dimensions

We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in $\mathbb{R}^2 $ is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. More precisely, we establish a quantitative estimate on how "close" these sets are (in the $L^{1}$ -sense and in the $L^{\infty}$ -sense) to be a halfplane, depending on $R$. This is a joint work with Joaquim Serra and Enrico Valdinoci.