Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter

In this paper, we consider the asymptotic behavior of the fractional mean curvature when $s\to 0^{+}$. Moreover, we deal with the behavior of $s$-minimal surfaces when the fractional parameter $s\in (0,1)$ is small, in a bounded and connected open set with $C^2$ boundary $\mathrm{\Omega }\subset {\mathbb{R}}^n$. We classify the behavior of $s$-minimal surfaces with respect to the fixed exterior data (i.e. the $s$-minimal set fixed outside of $\mathrm{\Omega }$). So, for $s$ small and depending on the data at infinity, the $s$-minimal set can be either empty in $\mathrm{\Omega }$, fill all $\mathrm{\Omega }$, or possibly develop a wildly oscillating boundary. Also, we prove the continuity of the fractional mean curvature in all variables, for $s\in (0,1]$. Using this, we see that as the parameter $s$ varies, the fractional mean curvature may change sign.