On the shape of hypersurfaces with boundary which have zero fractional mean curvature

We consider hypersurfaces with boundary in $\mathbb{R}^N$ that are the critical points of the fractional area introduced by Paroni, Podio-Guidugli, and Seguin in [R. Paroni, P. Podio-Guidugli, B. Seguin, 2018]. In particular, we study the shape of such hypersurfaces in several simple settings. First, we show that the critical points whose boundary is an $(N-2)$-sphere coincide with $(N-1)$-balls. Second, we show that the critical points whose boundary is the union of two parallel $(N-2)$-spheres do not coincide with two parallel $(N-1)$-balls. Moreover, the interior of the critical points does not intersect the boundary of the convex hull of the two $(N-2)$-spheres, while it can happen in the situation considered by Dipierro, Onoue, and Valdinoci in [S. Dipierro, F. Onoue, E. Valdinoci, 2022]. We also obtain a quantitative bound which may tell us how different the critical points are from the two $(N-1)$-balls. Finally, in the same setting as in the second case, we show that, if the two parallel boundaries are far away from each other, then the critical points are disconnected and, if the two parallel boundaries are close to each other, then the boundaries are in the same connected component of the critical points when $N \geq 3$. Moreover, by computing the fractional mean curvature of a cone with the same boundaries as those of the critical points, 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.


Introduction
Fractional minimal surfaces without boundary were first investigated by Caffarelli, Roquejoffre, and Savin in [6] and, since then, this topic has attracted many authors to study their geometric properties as an analogy of classical minimal surfaces.Roughly speaking, a fractional (or nonlocal) minimal surface without boundary is given as the boundary of a set which minimizes an energy functional defined by the pointwise interaction of a set and its complement.The typical interaction taken into account is scaling and translation invariant with some polynomial decay.Precisely, if s ∈ (0, 1) and Ω is an open set with smooth boundary, one of such standard energies of a set E ⊂ R N relative to Ω is the so-called fractional perimeter in Ω and is defined by where we denote by E c the complement of E. With this notion, we say that a set E ⊂ R N is a minimizer of P s relative to Ω if it holds that for any open bounded set Ω ′ ⊂ Ω and any F ⊂ R N with F \ Ω ′ = E \ Ω ′ .The existence and regularity of such minimizers was shown by Caffarelli, Roquejoffre, and Savin in [6].Moreover, they showed in [6] that, if a set E ⊂ R N is a minimizer of P s , then the following Euler-Lagrange equation holds in the viscosity sense: 2) is the so-called fractional mean curvature on the boundary ∂E.Dipierro, Savin, and Valdinoci in particular have revealed many properties which classical minimal surfaces cannot possess (see, for instance, [16,17] for the detail).In addition, many authors have studied the fractional(nonlocal) minimal surfaces or minimal graphs for more than a decade since the fractional(nonlocal) minimal surfaces appear in many other topics in which a long-range interaction is involved (see [11,30]).For further discussions about the geometric features of fractional(nonlocal) minimal surfaces without boundary, we refer to [2, 4, 5, 7, 9, 12-15, 18, 19].
Quite recently, motivated by some mathematical modelling of thin elastic structures, Paroni, Podio-Guidugli, and Seguin in [27] introduced a new notion of fractional areas and fractional mean curvatures for smooth manifolds which are not necessarily closed in the following way: let Ω ⊂ R N be a bounded domain and let M ⊂ Ω be any (N − 1)dimensional compact smooth manifold with or without boundary.Then the fractional area of M relative to Ω is defined by Per s (M; Ω) := c N X (M) max{χ Ω (x), χ Ω (y)} |x − y| N +s dx dy (1.3) where c N is some positive dimensional constant and X (M) is a set of all pairs (x, y) ∈ R N × R N such that the segment [x; y] with two end points x and y has an odd number of cross intersections with M and [x; y] is not tangent to M. Note that the presence of the term max{χ Ω (x), χ Ω (y)} in (1.3) is necessary to ensure that the integral converges whenever ∂M = ∅.
This manuscript is devoted to develop the theory of the fractional area Per s for manifolds with boundary.In particular, we aim to investigate the shape and topology of critical points of Per s .Here the critical point of Per s is defined by a smooth manifold such that the first variation of Per s vanishes with respect to a perturbation associated with the unit normal vector of that manifold (in the sequel, we will call this perturbation "normal variations").The authors in [27] obtained a necessary and sufficient condition for the vanishing of the first variation for manifolds as follows: let M be an orientable compact smooth manifold with or without boundary and assume that M is contained in a bounded domain Ω ⊂ R N .Then it holds that Here we denote by δPer s (M; Ω) the first variation of M under normal variations and H M,s is the fractional mean curvature associated with Per s which is defined by |y − z| N +s dy for any z ∈ M where c N is as in (1.3) and the sets A i (z) and A e (z) are defined by The sets A i (z) and A e (z) can be regarded as the "interior" and "exterior" of M relative to the point z, respectively, and these sets are determined uniquely once the unit normal vector of M at z is specified.See [27] for more discussions on the notions.Note that, if a manifold is not orientable, then the unit normal vector of the manifold cannot be determined uniquely and neither can the "interior" A i and "exterior" A e .Moreover, in this paper, we require the C 1,α -regularity with α > s of hypersurfaces so that the fractional mean curvatures are finite everywhere.
The study of critical points or fractional minimal surfaces with boundary can be related to the classical problem on free boundary minimal surfaces in differential geometry.One of the main topics in the problem is to determine the shape of a manifold Σ (embedded or immersed) in another smooth manifold S such that Σ minimizes its area in S and ∂Σ ⊂ ∂S with some topological constraints.The study of this classical problem was first considered by R. Courant in [10] in 1940 and, since then, a lot of authors have been intensively working on this topic.See, for instance, [22,24,26,29,32] for the detail.We also refer the readers to two surveys: [20] for classical works and [23] for more recent results.The references here are obviously not exhausted.
As an analogy of the classical free boundary minimal surfaces, it is natural to consider a fractional(nonlocal) version of free boundary minimal surfaces; however, the nonlocal version is not understood so far because, to our knowledge, suitable notions of fractional areas for manifolds with boundary had not been considered until Paroni, Podio-Guidugli, and Seguin in [27] introduced the notion of Per s in (1.3).To tackle the nonlocal version of the free boundary minimal surface problem, it is important to understand the geometric properties of critical points of Per s .
Given the importance of critical points of Per s from the above perspective, it is desirable to develop some intuition about their geometric features.To do this, since it is quite difficult to have explicit solutions which entirely describe critical points or minimizers of Per s , it is often convenient to study simplified cases in which the boundary of the critical points has some special characteristics.In this paper, we basically consider three cases: the first is that the boundary of critical points in R N lies in a hyperplane and is homeomorphic to S N −2 (our result is also true if the boundary is not always homeomorphic to S N −2 ).The second is that the boundary is the union of two distinct parallel and co-axial manifolds each of which lies in a hyperplane, is homeomorphic to S N −2 , and has distance of d from another boundary.The last is that the distance d is sufficiently large or sufficiently small.
Our first goal in this paper is to determine the shape of critical points of Per s whose boundary lies on a hyperplane.Precisely, we first define a set C ⊂ R N as where G is a non-empty bounded open subset of {x N = 0} with a smooth boundary.
Then we define an (N − 2)-dimensional smooth manifold Γ as Assume that M ⊂ R N is an orientable compact (N − 1)-dimensional C 1,α manifold with ∂M = Γ 0 and that M is a critical point of Per s .Note that the orientability of M implies the orientability of ∂M = Γ 0 .Then, as our first theorem, we aim to rigorously prove that M must coincide with C ∩ {x N = 0}, as we can intuitively expect this to be true.
Theorem 1.1.Let s ∈ (0, 1).Let Γ 0 be as in (1.8).Let M be an orientable compact Our second goal in this paper is to study the shape of critical points of Per s whose boundary consists of two disjoint components.The problem setting in the second theorem is as follows: we define two distinct compact (N − 2)-dimensional smooth manifolds Γ 1 and Γ 2 by where C is as in (1.7) and h 1 and h 2 are given constants with h 2 < h 1 .Then a critical point exhibit a different shape from a hyperplane.Precisely we prove Theorem 1.2.Let s ∈ (0, 1).Let Γ 1 and Γ 2 be as in (1.9) and let M be an orientable compact (N −1)-dimensional C In particular, We remark that, by using a cone whose boundary is Γ 1 ∪ Γ 2 as in Theorem 1.2 with h 1 = 1 and h 2 = −1, we can further detect how the critical points behave.See Subsection 3.2 of Section 3 for the detail.
Our third goal is to further study the shape and, in particular, the topology of critical points of Per s in the same situation as the one in Theorem 1.2.Precisely, taking Γ 1 and Γ 2 as in Theorem 1.2 with d := h 1 − h 2 > 0, we will see what critical points of Per s under normal variations look like in terms of connectedness if d is sufficiently large or sufficiently small.
To reach the third goal, we first show the following lemma which somehow tells us how different critical points are from hyperplanes.
Lemma 1.3.Let s ∈ (0, 1) and d > 0. Let Γ 1 and Γ 2 be as in (1.9) with h 1 = 0 and h 2 = −d.Assume that C is convex where C is as in (1.7).Then there exists a constant ε 0 > 0, depending only on N, s, and d, such that the following holds: let M be an orientable compact To favor the intuition, a sketch of our critical points is given in Figure 1.As a result of Lemma 1.3, we prove that, if the distance d between two parallel and co-axial boundaries is sufficiently small, then any critical point is connected in the sense that the two boundaries are in the same connected component when N ≥ 3.Moreover, when N = 2, any critical point is disconnected and its two distinct connected components should look like the right-hand side of Figure 1 with 0 < d ≪ 1.
Precisely, our third theorem is as follows.
Theorem 1.4.Let s ∈ (0, 1).Let Γ 1 and Γ 2 be as in Lemma 1.3.Assume that C is convex where C is as in (1.7).Then there exists d 0 = d 0 (N, s) > 0 such that the following holds: for any d ∈ (0, d 0 ), we take any orientable compact Moreover, when N = 2, there exist two distinct connected components M 1 and M 2 of M such that dist (M 1 , M 2 ) ≥ c with some constant c > 0, depending only on N and s, and ∂M i intersects both Γ 1 and Γ 2 for each i ∈ {1, 2}.
As a counterpart of Theorem 1.4, we prove that, if the distance d between two parallel and co-axial boundaries is sufficiently large, then any critical point is disconnected in any dimensions and it should look like the left-hand side of Figure 1 with d ≫ 1.
Our last theorem is as follows.
Theorem 1.5.Let s ∈ (0, 1).Let Γ 1 and Γ 2 be as in Lemma 1.3.Assume that C is convex where C is as in (1.7).Then there exists d 1 = d 1 (N, s) > 0 such that the following holds: we assume that, for any The topological properties in Theorem 1.4 and 1.5 could be expected to be true because Dipierro, Valdinoci, and the author of this paper obtained similar results in [14] on the topology of fractional minimal surfaces without boundary in the similar situations.On one hand, they showed that minimizers of P s in a given cylinder coincides with the cylinder itself for sufficiently small d where d is the distance between two disjoint parallel and co-axial external(boundary) data.On the other hand, they showed that minimizers of P s in the cylinder are disconnected for sufficiently large d.
Interestingly, however, we show in Theorem 1.2 that the critical points (not necessarily fractional area-minimizing) cannot touch the boundary of the cylinder C no mater what distance two parallel and co-axial boundaries have, while it is shown in [14] that minimizers of P s in a cylinder favorably stick to the boundary of the cylinder if N = 2 and d is large or if N ≥ 2 and d is small.Moreover, our results together with Remark 4.1 of Section 4 possibly indicate that critical points of Per s with two nearby parallel and co-axial compact boundaries might develop necks of catenoids, while this is not the case with fractional minimal surfaces considered in [14].We remark that the existence of fractional minimal catenoids without boundary in R 3 was shown by Dávila, Del Pino, and Wei in [13] if s is close to 1.
The organization of this paper is as follows: in Section 2, we prove Theorem 1.1 by "sliding" a hyperplane until it touches critical points (see the proof of Theorem 1.1 for the detail).In Section 3, we first give the proof of Theorem 1.2 and then we study further properties of critical points of Per s , computing the fractional mean curvature of a cone passing through the boundary of critical points.In Section 4, we first give the proof of Lemma 1.3 by constructing a suitable barrier and then, by using this lemma, we prove Theorem 1.4.Moreover, in Section 4, we also prove Theorem 1.5 by means of the "sliding method" (see Section 4 for the detail).

Proof of Theorem 1.1
In this section, we prove Theorem 1.1.The idea of the proof is inspired by the so-called sliding method introduced by Dipierro, Savin, and Valdinoci in [16].They developed this method in order to investigate the shape of fractional(nonlocal) minimal surfaces (see also [14,15,17] for further discussions).
We proceeds with the proof in the following way: we slide a hyperplane, parallel to C ∩ {x N = 0}, from below or above until it touches M and assume by contradiction that there exists a touching point in (C ∩ {x N = 0}) c .At the touching point q, we obtain the Euler-Lagrange equation (1.4).Then, taking into account all the contributions from the "interior" A i (q) and the "exterior" A e (q) of M, we can observe that the contribution from either A i (q) or A e (q) turns out to be strictly larger than that from the other region.This contradicts the Euler-Lagrange equation.
Proof of Theorem 1.1.We first define a hyperplane H λ := {(x ′ , x N ) | x N = λ} and two half-spaces for λ ∈ R. We set P λ : R N → R N as the reflection map with respect to H λ for λ ∈ R and set x λ := P λ (x) for any x ∈ R N .Moreover, we denote by C Γ 0 (q) a (filled) cone with vertex q whose boundary passes through Γ 0 , that is, We further set C λ Γ 0 (q) := P λ (C Γ 0 (q)).Now let M ⊂ R N be the critical point chosen in Theorem 1.1.The minimizer M is bounded.Hence, we can slide the hyperplane H λ from below until it touches the minimizer M. Our result in Theorem 1.1 states that this touching does not occur in H − 0 ∪ H + 0 and thus, we assume by contradiction that there exist a constant λ 0 < 0 and a point q ∈ M ∩ Ω such that where T q M is a tangent space of M at q. Due to the symmetry of our setting, we can conduct the same argument that we will show below in the case that we slide the hyperplane from above and the touching occurs in H + 0 .Hence, it is sufficient to show the proof in the case that the touching occurs in H − 0 .See also Figure 2 for the situation that we consider in dimension 2.
Since M is an orientable compact critical point of Per s , which means the vanishing of the first variation of Per s at M, and since q ∈ M, we obtain, from (1.4), that where the sets A e (q) and A i (q) are defined as in (1.5) and (1.6).We consider all the contributions from A e (q) and A i (q) in detail and show that the singular integral in the right-hand side of (2.2) is strictly positive, which is a contradiction.Indeed, since C Γ 0 (q) ⊂ H + λ 0 and H λ 0 is tangential to M, we have that P λ 0 (A e (q)) ⊂ H − λ 0 ⊂ A i (q).This implies that R N = A e (q) ∪ P λ 0 (A e (q)) ∪ A i (q) \ P λ 0 (A e (q)), up to negligible sets, and thus we can compute the fractional mean curvature H M,s at q as follows: From the change of variables y → P λ 0 (y) and the definition of P λ 0 , we have (2.4) Moreover, we have that the volume of the set A i (q) \ P λ 0 (A e (q)) is not zero because See also Figure 3 for illustration in dimension 2. From (2.3) and (2.4), we obtain which is a contradiction.Figure 3: The same situation as in Figure 2. The reflection P λ 0 (A e (q)) of A e (q) is shown in dark gray, the set A e (q) in light gray.

Shape of Critical Points with Two Disjoint Compact Boundaries
In this section, we first give the proof of Theorem 1.2 and then we further show some properties of critical points of Per s and compute the fractional mean curvature of cones.

Proof of Theorem 1.2
In this subsection, we prove Theorem 1.2.The idea of the proof is basically the same as the one in the proof of Theorem 1.1.The convexity assumption on C is necessary for us to use the sliding method.
Proof of Theorem 1.2.We first define Let M ⊂ R N be the critical point chosen in Theorem 1.2.By using the same argument as we show in the proof of Theorem 1.1, we obtain that M cannot exist in the regions We now show that any connected component of M cannot be either C 1 or C 2 .To see this, we assume by contradiction that there exists a connected component M 1 of M such that M 1 coincides with C 1 .Taking any q ∈ M 1 , we have that the cone C q,Γ 2 of vertex q whose boundary passes through Γ 2 is contained in H − Γ 1 .By choosing a proper orientation of M, we can have that H + Γ 1 ⊂ A e (q) and A i (q) ⊂ H − Γ 1 where the sets A e (q) and A i (q) are defined as in (1.5) and (1.6), respectively.See Figure 4 for the situation in dimension 2. Since M is a critical point of Per s , from (1.4), we have that |y − q| N +s dy.
Now, by employing the same argument we show in the proof of Theorem 1.1, we obtain that This contradicts (3.1).Therefore, we conclude that the first claim is valid.
To prove the rest of the claim, we can argue in the same way as in the proof of the first claim.Indeed, we slide any hyperplane parallel to the x N -axis from right to left or from left to right until it touches the boundary of the cylinder C. If there is no touching point, from the convexity of C, we obtain that the critical point M is strictly contained in C except for its boundary.Thus, we assume by contradiction that there exists a touching point q of M in the complement of C.Then, by choosing a proper orientation of M, we can show that the contribution from A e (q) relative to the touching point q is strictly larger (or smaller) than that from A i (q), respectively, as we see in the proof of the first claim.This contradicts that the fractional mean curvature vanishes at the touching point q.Therefore, we conclude the proof of Theorem 1.2.

Further Study on Critical Points and Cones
In this subsection, we study more the shape of critical points of Per s in the same situation as in Theorem 1.2 with h 1 = 1 and h 2 = −1.
First, we investigate the shape of critical points in dimension 2. To see this, we divide R 2 into four regions, that is, we define four regions C t 0 , C b 0 , C r 0 , and C ℓ 0 by respectively.Moreover, we set Notice that ∂C 0 = Γ 1 ∪ Γ 2 where Γ 1 and Γ 2 are given in Theorem 1.2 with h 1 = 1 and From the definition of Γ 1 and Γ 2 , we have that Γ 1 = {(±1, 1)} and Γ 2 = {(±1, −1)}.Now we prove that the fractional mean curvature of the cone C 0 vanishes at regular points, i.e., H C 0 ,s (z) = 0 ( for any z ∈ C 0 \ ∂C d with z = 0. Indeed, let z ∈ C 0 \ {0, (±1, 1), (±1, −1)} and, by symmetry, we may assume that z = (z 1 , z 2 ) satisfies −1 < z 1 < 0 and 0 < z 2 < 1.Then, from the definition of the "interior" A i (z) and the "exterior" A e (z) of the cone C 0 and by taking a suitable orientation of C 0 \ {0}, we may obtain that where we denote by [p, q] the straight line passing through p, q ∈ R 2 with p = q and we define [p, q] + and [p, q] − by the upper part and the lower part of the region separated by the straight line [p, q], respectively.Now, because of the symmetry of the cone C 0 , we readily observe that, in dimension 2, the sets A i (z) and A e (z) are equivalent to each other in the sense that By definition, we notice that T (z) = z.
Therefore, from the change of variables x → T (x) and , we obtain that By combining this fact with Theorem 1.2, we can prove the following proposition.

7). If γ is a critical point of Per s under normal variations, then γ is not contained in either C
Remark 3.2.We may observe, by combining Proposition 3.1 with Theorem 1.2, that the possible shape of minimizers of Per s in dimension 2 whose boundary is Γ 1 ∪ Γ 2 is depicted in Figure 5. Now, by choosing a proper orientation, we consider the "interior" and "exterior" of γ and C 0 at the touching point z.To see this, we set the interior and exterior at q ∈ η of a curve η ⊂ R 2 as A η i (z) and A η e (z), respectively.Then, from the smoothness of the critical point γ and the assumption that γ ⊂ C t 0 ∪ C b 0 , we obtain, by taking a suitable orientation of γ and C 0 , that Here, from Theorem 1.2, we have used the fact that all the critical points of Per s in our situation are contained in the box See also Figure 6 for our situation.Hence, since γ is a critical point of Per s , we have that   We next prove the same result as Proposition 3.1 in higher dimensions.To see this, we also show that the fractional mean curvature of a cone passing through Γ 1 ∪ Γ 2 is either positive or negative everywhere except at its vertex in higher dimensions.The idea of the proof is the same as that in the proof of Proposition 3.1.We first give some notations.We define a bounded tube D 0 and a unbounded cone C 0 by and decompose C 0 into two parts C + 0 and C − 0 which are defined by Proposition 3.4.Let N ≥ 3 and s ∈ (0, 1).Let Γ 1 and Γ 2 be as in Theorem 1.2 with h 1 = 1 and Proof.The proof is similar to that of Proposition 3.1 and we here show a rough sketch of the proof.Let M be the critical point selected in Proposition 3.4.We assume that (M \ ∂M) ∩ (C N 0 \ {0}) is not empty and we choose a point z ∈ (M \ ∂M) ∩ (C N 0 \ {0}).Suppose by contradiction that either holds.First, by choosing a proper orientation, we show that Indeed, if we take the unit normal vector ν C N 0 (z) of the cone C N 0 at z in such a way that the direction is towards C 0 , then the "interior" A C N 0 i (z) and "exterior" A C N 0 e (z) can be defined as where C Γ i (z) is defined by a (filled) cone of vertex z passing through Γ i for each i ∈ {1, 2}.Now we take a hyperplane H z which is tangent to ∂ C 0 and passes through z and define the reflection map T Hz with respect to H z .From the definitions of C N 0 , A and  A C N 0 e (z), we have Since T Hz is an isometry and T Hz (z) = z, we obtain the following: which implies (3.8).Now, since M is a critical point of Per s , we have the Euler-Lagrange equation Thus, taking the unit normal vector ν M (z) of M at z as ν C N 0 (z), we can have the following computation: From the assumption, we can observe that Therefore, from (3.8) and (3.10), we reach a contradiction.

Topology of Critical Points
In this section, we investigate the topology of critical points with two parallel and co-axial boundaries and prove Theorem 1.4 and 1.5.Before proving our main theorems of this section, we show Lemma 1.3.The idea of the proof is to construct a small barrier, whose fractional mean curvature is strictly positive or negative, and to "slide" the barrier until it touches the critical point.The construction of the barrier is inspired by the one shown in [18].See also [14, Proof of Proposition 4.1].In the sequel, without loss of generality, we may assume that where C is as in (1.7) for simplicity.
Proof of Lemma 1.3.We first fix ε ∈ (0, 1) so small that δ = δ(ε) := (− log ε) −1/2 < 1 2 and we define a smooth bump function w ε : R N −1 → R by If necessary, we may choose ε in such a way that φ(ε) < 1.Note that, since φ is an increasing function in a neighborhood I φ ⊂ [0, 1) of the origin, its inverse function φ −1 exists in a neighborhood J φ ⊂ [0, 1) of the origin.We then set r(ε) := (2(N − 1)φ(ε)) −1 and d(ε) := 2r(ε).( Moreover, we define a positive constant ε d as By definition, we observe that r(ε d ) ≥ d and ε d can be chosen independently of In addition, we choose a smooth function v ε : where we denote by B ′ r (0) an open ball centered at the origin of radius r in R N −1 .In particular, we choose v ε in such a way that its subgraph with the base of radius 1  16 .Then we define a function w ε : R N −1 → R by Notice that w ε is smooth in R N −1 .Now we construct a barrier against M ε,t , i.e., an orientable compact (N−1)-dimensional piecewise smooth manifold M ε,t in the following way: first, taking any t ∈ (0, ε], we define four sets where d(ε) := 2r(ε).Then we define our barrier as M ε,t := M ε,t 1 ∪M ε,t 2 .By construction, we can easily see that M ε,t is an orientable compact (N −1)-dimensional smooth manifold with ∂M ε 1 = Γ ε,t 1 and ∂M ε,t 2 = Γ ε,t 2 where we define We next construct another barrier in which the small bump associated with v ε is removed from M ε,t .First, for any t ∈ (0, ε], we define a manifold M ε,t 3 as the graph of w ε , i.e., M ε,t ) + t} and, then, define the second barrier as M ε,t := M ε,t 3 ∪M ε,t 2 .Notice that ∂M ε,t = Γ ε,t 1 ∪Γ ε,t 2 .We now show, up to orientation, that the fractional mean curvature of M ε,t is negative on the graph of w ε .Let q ∈ M ε,t 1 be any point such that |q ′ | < δ(ε) where we set q = (q ′ , q N ).We now define C Γ ε,t i (q) by a (filled) cone of vertex q whose boundary passes Figure 7: The barrier M ε,t = M ε,t 1 ∪ M ε,t 2 associated with a function w ε in dimension 2. The graph of w ε in {|x ′ | < 1} is depicted with black lines and the cylinder in dark gray.through Γ ε,t i for i ∈ {1, 2}.Then, up to orientation, we can choose the interior and exterior of M ε,t at q as respectively, where we define a function w q ε : R N −1 → R by We now compute the fractional mean curvature H M ε,t ,s (q) at q of M ε,t .From the definition of the fractional mean curvature and by the change of variables, we have where we set r := r(ε) where r(ε) is as in (4.2).We first compute (I).Thanks to the choice of r and the construction of M ε,t , we observe that Thus we can represent the set ∂A M ε,t e (q) in B ′ r (0) × (−r, r) as the graph of w q ε .By doing the similar computation to the one in [3, Section 3], we obtain where we set N+s 2 dσ for any t ∈ R. Note that we have used the change of variables x ′ → −x ′ in the second equality of (4.4) and the fact that F is odd in the last equality of (4.4).By definition, we have that w q ε (q ′ ) = w ε (q ′ ) and w q ε ≥ w ε in R N −1 .Since F is increasing, we derive from (4.4) that Now, by using the fundamental theorem of calculus in (4.4), we obtain where we set a(x ′ , q ′ , λ) as . Thus, by using again the fundamental theorem of calculus, we have Since w ε is smooth in R N −1 , we then have Now we compute (II) in the following way: since B r (0 Therefore, from (4.7) and (4.8), we obtain where c 1 and c 2 are defined as respectively.From (4.2), it holds that the right-hand side of (4.9) takes the maximum at r = r(ε) ∈ (0, d(ε)).Hence we finally obtain, from (4.9), that where we set the constant c = c(N, s) > 0 as Next we compute the fractional mean curvature H M ε,t ,s (q) at q by using Estimate (4.10) of the fractional mean curvature H M ε,t ,s (q) at q. Indeed, from the construction of M ε,t and M ε,t , we have that, by choosing a proper orientation, A M ε,t e (q) ⊂ A M ε,t e (q) and thus we obtain )} and the subgraph contains the cylinder of height φ(ε) β with the base of radius 1/16, we have where a constant c ′ = c ′ (N) > 0 depends only on N.Moreover, we observe that the distance between q and the cylinder is less than, at most, 2 + φ(ε) β and this is bounded from above by some constant depending only on N, s, and β.Hence from (4.10) and (4.11) and by recalling the choice of φ, we obtain where c ′′ > 0 is a constant depending only on N, s, and β.Since 0 < β < s and φ(ε) ↓ 0 as ε ↓ 0, we choose ε 1 = ε 1 (N, s, β) ∈ I φ ∩ (0, 1 100 ) so small that the right-hand side of (4.12) is positive for any ε ∈ (0, ε 1 ].Therefore, from (4.12), we obtain that H M ε,t ,s (q) < 0 for ε ∈ (0, ε 1 ].Now we set ε 2 := min{ε 1 , ε d }.Since r(ε d ) ≥ d and δ(ε 2 ) < 1 2 , we may observe that d(ε 2 ) ≥ d and M ε 2 ,t 3 ∩ Γ 1 = ∅ for any t ∈ (0, ε 2 ].For our convenience, we denote ε 2 by ε in the sequel.
We then slide the barrier M ε,t from above, i.e., we vary the parameter t stating at ε until M ε,t touches the critical point M. To prove the claim, we assume by contradiction that there exists t 1 ∈ (0, ε] such that M∩M ε,t 1 3 = ∅ and M∩M ε,t 3 = ∅ for any t ∈ (t 1 , ε].We pick up a point q ε,t 1 ∈ M ∩ M ε,t 1  3 .Notice that See Figure 8 to favor the intuition in dimension 2. Figure 8: The critical point γ depicted with dashed lines and the barrier γ ε,t 1 with black line.γ touches γ ε,t 1 at q ε,t 1 from above.The exterior A γ ε,t 1 e (q ε,t 1 ) of γ ε,t 1 is depicted in light gray and the exterior A γ e (q ε,t 1 ) of γ in both light and dark gray.
Since M is a critical point of Per s under normal variations, we obtain H M,s (q ε,t 1 ) = 0.
From Theorem 1.2 and the above argument, we obtain that the touching point q ε,t and thus H M ε,t 1 ,s (q ε,t 1 ) < 0.Moreover, from the construction of M ε,t 1 1 , we have, by choosing a proper orientation, that From the symmetry of the balls, we have (q) where C 2 := C 0 + C 1 is a constant depending only on N and s.From (4.16) and (4.17), we have |x − q| N +s dx.(4.18) Now we consider the contributions from A M i (q) and A M e (q) in S c d .We now define C Γ (q) by a (filled) cone of vertex q whose boundary passes through Γ. Moreover we define C S d (q) by a (filled) cone of vertex q whose boundary passes through From the definitions of S d and Γ, we observe that C Γ (q) ⊂ C S d (q).We now set C Γ (q) := C Γ (q) ∩ {(x ′ , x N ) | x N > 0 or x N < −d}.We then rotate C Γ (q) by angle π/2 or −π/2 with respect to the straight line parallel to the x 1 -axis passing trough q (if N = 2, then we just rotate C Γ (q) by angle π/2 or −π/2 with respect to q).Since we choose d so large that d − d α > 100, we obtain that where R( C Γ (q)) is an image of C Γ (q) by the rotation map R : R N → R N in the above.See Figure 10 for an intuitive understanding.Then, observing that R(q) = q and S c d ∩ A M e (q) = C Γ (q) and by a change of variables, we have Figure 10: The touching between the ball B d α /2 (p t 0 ,d ) and its symmetric ball B at q.The image of the set C Γ (q) by the rotation map R is depicted in dark gray and the set A γ e (q) in light gray.
From the definitions of S d and the rotation map R, we can choose an open ball outside S d and R( C Γ (q)) but close to q, i.e., we have where we set e 1 := (1, 0, • • • , 0) ∈ R N .Thus, we obtain

χ
E c (y) − χ E (y) |y − x| N +s dy = 0 (1.2) for x ∈ ∂E.The integral in (1.2) is intended in the Cauchy principal value sense.This can be regarded as a nonlocal counterpart of the classical minimal surface equation and the left-hand side in (1. Per s under normal variations, then a set enclosed by M and the union of C ∩ {x N = 0} and C ∩ {x N = −d} contains two half-balls B − ε 0 (0) := {x ∈ B ε 0 (0) | x N < 0} and B + ε 0 (p d ) := {x ∈ B ε 0 (p d ) | x N > −d} where p d := (0, −d) ∈ R N −1 × R.

Figure 1 :
Figure 1: Two possible situations in dimension 2 in Theorem 1.3 in which the 'interior" or "exterior" of the critical point γ with ∂γ = Γ 1 ∪ Γ 2 contains two half-balls.

Figure 2 :
Figure 2: The situation in dimension 2 in which the critical point M = γ is a C 1,α curve with ∂γ = {±a} where ±a := (±a, 0).The set A e (q) is shown in dark gray, the set A i (q) in white.The dashed lines represent the boundary of the cone C ±a (q).

Figure 4 :
Figure 4: The situation in dimension 2 in which each componentM i = γ i of the critical point M = γ for i ∈ {1, 2} is a C 1,α curve with ∂γ i = Γ i where Γ 1 = {±a} and Γ 2 = {±b}.The set A e (q) is shown in gray, the set A i (q) in white.

Figure 5 :
Figure 5: Possible minimizers γ of Per s in dimension 2 with ∂γ = Γ 1 ∪ Γ 2 is shown with dashed lines.On the right, γ does not intersect with C 0 except at their boundaries Γ 1 and Γ 2 .

Figure 6 :
Figure 6: The situation of the critical point γ and the touching point z in which γ is included in C t 0 ∪ C b 0 with ∂γ = Γ 1 ∪ Γ 2 .The set A γ e (z) is shown in light gray, the set A C 0 i (z) in white, and the set A C 0 e (z) \ A γ e (z) in dark gray.

eRemark 3 . 3 .
(z)\A γ e (z) 1 |y − z| 2+s dy > 0, (3.6) which is a contradiction.Therefore we obtain the claim.We briefly consider the situation of Theorem 1.2 with h 1 = d and h 2 = −d for d = 1 and d > 0 and see what kind of shape the critical points in dimension 2 look like.Notice that we have treated the case of d = 1 in Proposition 3.1.Assume that h 1 = d and h 2 = −d for d > 0. We define a cone C d of vertex 0 by

. 7 )
Notice that ∂C d = Γ 1 ∪Γ 2 .By slightly modifying the argument for showing that H C 0 ,s = 0 on C 0 \(∂C 0 ∪{0}) and taking a proper orientation, we can show that the fractional mean curvature H C d ,s (z) of C d is either positive or negative for any z ∈ C d \ ∂C d with z = 0.Then, again by slightly modifying the argument in the proof of Proposition 3.1, we obtain the same result as in Proposition 3.1 even for any d = 1.
1,α manifold with ∂M = Γ 1 ∪Γ 2 .Assume that C is convex where C is as in (1.7).If M is a critical point of Per s under normal variations, then any connected component of M is neither C 1 nor C 2 where we define