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

Abstract

AbstractWe consider compact hypersurfaces with boundary in $${\mathbb {R}}^N$$ 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 $$\Gamma $$ Γ of dimension $$N-2$$ N - 2 and lies in a hyperplane $$H \subset {\mathbb {R}}^N$$ H ⊂ R N . Then we show that the critical points coincide with a smooth manifold $${\mathcal {N}}\subset H$$ N ⊂ H of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}= \Gamma $$ ∂ N = Γ . Second we consider the critical points whose boundary consists of two smooth, orientable, closed manifolds $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 of dimension $$N-2$$ N - 2 and suppose that $$\Gamma _1$$ Γ 1 lies in a hyperplane H perpendicular to the $$x_N$$ x N -axis and that $$\Gamma _2 = \Gamma _1 + d \, e_N$$ Γ 2 = Γ 1 + d e N ($$d >0$$ d > 0 and $$e_N = (0,\cdots ,0,1) \in {\mathbb {R}}^N$$ e N = ( 0 , ⋯ , 0 , 1 ) ∈ R N ). Then, assuming that $$\Gamma _1$$ Γ 1 has a non-negative mean curvature, we show that the critical points do not coincide with the union of two smooth manifolds $${\mathcal {N}}_1 \subset H$$ N 1 ⊂ H and $${\mathcal {N}}_2 \subset H + d \, e_N$$ N 2 ⊂ H + d e N of dimension $$N-1$$ N - 1 with $$\partial {\mathcal {N}}_i = \Gamma _i$$ ∂ N i = Γ i for $$i \in \{1,2\}$$ i ∈ { 1 , 2 } . Moreover, the interior of the critical points does not intersect the boundary of the convex hull in $${\mathbb {R}}^N$$ R N of $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ 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 $${\mathcal {N}}_1 \cup {\mathcal {N}}_2$$ 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 $$\Gamma _1$$ Γ 1 and $$\Gamma _2$$ Γ 2 are in the same connected component of the critical points when $$N \ge 3$$ N ≥ 3 . Moreover, by computing the fractional mean curvature of a cone whose boundary is $$\Gamma _1 \cup \Gamma _2$$ 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.