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

We consider compact hypersurfaces with boundary in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ that are the critical points of the fractional area introduced by Paroni et al. (Commun Pure Appl Anal 17:709–727, 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 \(\Gamma \) $\mathrm{\Gamma }$ of dimension \(N-2\) $N-2$ and lies in a hyperplane \(H \subset {\mathbb {R}}^N\) $H\subset {\mathbb{R}}^N$ . Then we show that the critical points coincide with a smooth manifold \({\mathcal {N}}\subset H\) $\mathcal{N}\subset H$ of dimension \(N-1\) $N-1$ with \(\partial {\mathcal {N}}= \Gamma \) $\partial \mathcal{N}=\mathrm{\Gamma }$ . Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ and \(\Gamma _2\) ${\mathrm{\Gamma }}_2$ of dimension \(N-2\) $N-2$ and suppose that \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ lies in a hyperplane H perpendicular to the \(x_N\) $x_N$ -axis and that \(\Gamma _2 = \Gamma _1 + d \, e_N\) ${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_1+d{e}_N$ ( \(d >0\) $d>0$ and \(e_N = (0,\cdots ,0,1) \in {\mathbb {R}}^N\) $e_N=(0,\cdots ,0,1)\in {\mathbb{R}}^N$ ). Then, assuming that \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds \({\mathcal {N}}_1 \subset H\) ${\mathcal{N}}_1\subset H$ and \({\mathcal {N}}_2 \subset H + d \, e_N\) ${\mathcal{N}}_2\subset H+d{e}_N$ of dimension \(N-1\) $N-1$ with \(\partial {\mathcal {N}}_i = \Gamma _i\) $\partial {\mathcal{N}}_i={\mathrm{\Gamma }}_i$ for \(i \in \{1,2\}\) $i\in \{1,2\}$ . Moreover, the interior of the critical points does not intersect the boundary of the convex hull in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ of \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ and \(\Gamma _2\) ${\mathrm{\Gamma }}_2$ , while this can occur in the codimension-one situation considered by Dipierro et al. (Proc Am Math Soc 150:2223–2237, 2022). We also obtain a quantitative bound which may tell us how different the critical points are from \({\mathcal {N}}_1 \cup {\mathcal {N}}_2\) ${\mathcal{N}}_1\cup {\mathcal{N}}_2$ . 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 \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ and \(\Gamma _2\) ${\mathrm{\Gamma }}_2$ are in the same connected component of the critical points when \(N \ge 3\) $N\ge 3$ . Moreover, by computing the fractional mean curvature of a cone whose boundary is \(\Gamma _1 \cup \Gamma _2\) ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2$ , 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.