\[NLMS\]. The red curve represents the graph of the function $f$ defined in .]{}]{}*[]{data-label="DOMA"}](traccia.pdf){width="8.9cm"}

The statement of Theorem \[NLMS\] is described[^1] in Figures \[DOMA\] and \[DOMA-2\]. With respect to this, we stress the remarkable geometric property given by the [*vanishing of the gradient of the trace at the zero crossing points*]{}. We observe that this situation is completely different with respect to the one arising for solutions of linear equations, and one can compare the structurally “rigid” geometry imposed by Theorem \[NLMS\] with the almost completely arbitrariness arising in Theorem \[FLFL\].

We remark that, at a formal level, the settings in Theorems \[FLFL\] and \[NLMS\] are strictly related, since the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem. More specifically, when one takes into account the improvement of flatness argument for an ${\varepsilon}$-flat nonlocal minimal graph $u$ (see the forthcoming Lemma \[NOANpierjfppp34\]), one sees that $u/{\varepsilon}$ shadows a function $\bar u$, which is a solution of $(-\Delta)^{\sigma}\bar u=0$ in $\{x_2>0\}$, with $\sigma:=\frac{1+s}2$: in this context the first order of $\bar u$ near the origin takes the form $\bar a \,x_2^{\sigma}$, for some $a\in{\ensuremath{\mathbb{R}}}$. Comparing with , one 