On the Shape of Hypersurfaces with Boundary Which Have Zero Fractional Mean Curvature

We consider compact hypersurfaces with boundary in $\mathbb{R}^N$ that are the critical points of the fractional area introduced by Paroni, Podio-Guidugli, and Seguin in 2018. In particular, we study the shape of such hypersurfaces in several simple settings. First we consider the critical points whose boundary is a smooth, orientable, closed manifold $\Gamma$ of dimension N − 2 and lies in a hyperplane $H \subset \mathbb{R}^N$. Then we show that the critical points coincide with a smooth manifold $\mathcal{N} \subset H$ of dimension N − 1 with $\partial \mathcal{N} = \Gamma$. Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds $\Gamma_1$ and $\Gamma_2$ of dimension N − 2 and suppose that $\Gamma_1$ lies in a hyperplane H perpendicular to the $x_N$-axis and that $\Gamma_2 = \Gamma_1 + d e_N$ ($d > 0$ and $e_N = (0, · · · , 0, 1) \in \mathbb{R}^N$). Then, assuming that $\Gamma_1$ has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds $\mathcal{N}_1 \subset H$ and $\mathcal{N}_2 \subset H + d \, e_N$ of dimension N −1 with $\partial N_i = \Gamma_i$ for $i \in \{1, 2\}$. Moreover, the interior of the critical points does not intersect the boundary of the convex hull in $\mathbb{R}^N$ of $\Gamma_1$ and $\Gamma_2$, while this can occur in the codimension-one situation considered by Dipierro, Onoue, and Valdinoci in 2022. We also obtain a quantitative bound which may tell us how different the critical points are from $\mathcal{N}_1 \cup \mathcal{N}_2$. Finally, in the same setting as in the second case, we show that, if d is sufficiently large, then the critical points are disconnected and, if d is sufficiently small, then $\Gamma_1$ and $\Gamma_2$ are in the same connected component of the critical points when $N \geq 3$. Moreover, by computing the fractional mean curvature of a cone whose boundary is $\Gamma_1 \cup \Gamma_2$, we also obtain that the interior of the critical points does not touch the cone if the critical points are contained in either the inside or the outside of the cone.