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 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 \(\Gamma \) 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 \(e_N = (0,\cdots ,0,1) \in {\mathbb {R}}^N\)). 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 \(\partial {\mathcal {N}}_i = \Gamma _i\) 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 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 \({\mathcal {N}}_1 \cup {\mathcal {N}}_2\). 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 \(N \ge 3\). 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.