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

We consider compact hypersurfaces with boundary in RN 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 Γ Γ of dimension $$N-2$$ N - 2 and lies in a hyperplane H ⊂ RN H ⊂ R N . Then we show that the critical points coincide with a smooth manifold N⊂ H N ⊂ H of dimension $$N-1$$ N - 1 with ∂ N= Γ ∂ N = Γ . Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds Γ ₁ Γ 1 and Γ ₂ Γ 2 of dimension $$N-2$$ N - 2 and suppose that Γ ₁ Γ 1 lies in a hyperplane H perpendicular to the xN x N -axis and that Γ ₂ = Γ ₁ + d \, eN Γ 2 = Γ 1 + d e N ( $$d >0$$ d > 0 and eN = (0,⋯ ,0,1) ∈ RN e N = ( 0 , ⋯ , 0 , 1 ) ∈ R N ). Then, assuming that Γ ₁ Γ 1 has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds N₁ ⊂ H N 1 ⊂ H and N₂ ⊂ H + d \, eN N 2 ⊂ H + d e N of dimension $$N-1$$ N - 1 with ∂ Nᵢ = Γ ᵢ ∂ N i = Γ i for i ∈ \1,2\ i ∈ { 1 , 2 } . Moreover, the interior of the critical points does not intersect the boundary of the convex hull in RN R N of Γ ₁ Γ 1 and Γ ₂ Γ 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 N₁ ∪ N₂ N 1 ∪ 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 Γ ₁ Γ 1 and Γ ₂ Γ 2 are in the same connected component of the critical points when N ≥ 3 N ≥ 3 . Moreover, by computing the fractional mean curvature of a cone whose boundary is Γ ₁ ∪ Γ ₂ Γ 1 ∪ Γ 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.