ON A NONLOCAL ISOPERIMETRIC PROBLEM ON THE TWO-SPHERE

In this article we analyze the minimization of a nonlocal isoperimetric problem (NLIP) posed on the 2-sphere. After establishing the regularity of the free boundary of minimizers, we characterize two critical points of the functional describing (NLIP): the single cap and the double cap. We show that when the parameter controlling the influence of the nonlocality is small, the single cap is not only stable but also is the global minimizer of (NLIP) for all values of the mass constraint. In other words, in this parameter regime, the global minimizer of the (NLIP) coincides with the global minimizer of the local isoperimetric problem on the 2-sphere. Furthermore, we show that in certain parameter regimes the double cap is an unstable critical point.


Introduction
There is currently much interest in the area of pattern formation for ordered structures on curved surfaces. Examples of this interest range from biology to material science: covering virus and radiolaria architecture, colloid encapsulation for possible drug delivery; and most relevant to this article, self-assembly in thin block copolymer melt films confined to the surface of a sphere (cf. [2] and references therein). There is an extensive literature on the mathematical analysis of phase separation of block copolymers and their sharp interface limit leading to a nonlocal isoperimetric problem, but to our knowledge, all of these investigations have been carried out on either the flat-tori or bounded domains in the Euclidean space (cf. [4,5,19,21,28] and references therein). Here, we have chosen to focus on the nonlocal isoperimetric problem on the two-sphere in order to present what is perhaps the first rigorous attempt to analyze a block copolymer related problem on a manifold with nonzero curvature.
We start by defining the nonlocal isoperimetric problem (NLIP) on the sphere: For fixed m ∈ (−1, 1) Here S 2 is the 2-sphere, ∆ denotes the Laplace-Beltrami operator on S 2 and H 2 denotes the 2-dimensional Hausdorff measure. Also, the first term in E γ is half of the total variation of u, which is defined by where X (S 2 ) denotes the set of all vector fields of class C ∞ on S 2 and divX refers to the divergence operator relative to S 2 , see e.g. [8,10]. Then, the first term in E γ indeed computes the perimeter of the set {x ∈ S 2 : u(x) = 1} since u takes on only values ±1 and the perimeter of a set Ω in S 2 is given by where χ Ω denotes the characteristic function of the set Ω. Finally, let us note that throughout this paper ∇ denotes the gradient relative to the sphere S 2 whereas ∇ ∂Ω denotes the gradient relative to the submanifold ∂Ω of a subset Ω ⊂ S 2 . The problem (NLIP) arises, up to a constant factor, as the Γ-limit as ε → 0 of the wellstudied Ohta-Kawasaki sequence of functionals E ε,γ which model microphase separation of diblock copolymers, [3,20]: where again v satisfies (1.2). There is an extensive literature exploring the energy landscape for E ε,γ in two and three dimensions, whether posed on the flat torus (i.e. with periodic boundary conditions) or on a general domain with homogeneous Neumann data, cf. e.g. [5,6,22,24] and references therein. The picture is quite rich and complicated, with the diffuse interface sometimes bounding one or more strips, wriggled strips, discs or ovals. Again though, we are not aware of work on Ohta-Kawasaki posed on S 2 . Independent of its connection to Ohta-Kawasaki, (NLIP) also attracts interest as a rather canonical nonlocal perturbation of the classical isoperimetric problem. Indeed, as a model for pattern formation, (NLIP) sets up a basic competition between low surface area (the perimeter term) and high oscillation (the nonlocal term) and much the same richness exists for the energy landscape of (NLIP) defined on the flat torus or on a general domain with homogeneous Neumann data. In three dimensions, computations reveal a wide array of stable critical points, with the free boundary ∂{x : u(x) = 1} consisting of one or more pairs of parallel planes, one or more spheres, cylinders or even hypersurfaces resembling more exotic triply periodic constant mean curvature surfaces such as gyroids, depending on where in the (m, γ) parameter space one looks. With few exceptions, however, rigorous proofs of stability for particular patterns are rare (cf. e.g. [4,25,26,27,28]).
Patterns emerging by phase separation of diblock copolymers have also been investigated numerically on spherical surfaces. In [33], the authors explore the phase separation on spherical surfaces by solving the time-dependent Cahn-Hilliard equation modified for diblock copolymers using a finite volume method, whereas in [2], the authors develop a numerical method for solving the self-consistent field theory equations in spherical geometries and address detailed numerical simulations of both lamellar and hexagonal ordering of a spherical thin film of diblock copolymer. Here we have chosen to focus on the problem (1.1) defined on the two-sphere S 2 with the hope of initiating a rigorous study of block copolymer models on curved surfaces.
This article represents a sequel to the recent work [30] on the characterization of global minimizers of (NLIP) on the flat two-torus. There we show that the minimizers are lamellar for an interval of m-values containing m = 0 when γ is sufficiently small. The main idea in [30] uses the Γ-convergence of E γ and relies on the fact that the lamellar critical point is the global minimizer of the (local) periodic isoperimetric problem for the same interval of m-values when γ = 0.
In the case of the two-sphere, when γ = 0, the problem reduces the (local) isoperimetric problem and it is known that the single spherical cap, i.e., the set with boundary consisting of a single circle, is the global minimizer for any m ∈ (−1, 1) (cf. [17,29]). In light of this, we are able to carefully adapt the arguments in [30] and show that, for γ sufficiently small, the single cap is not only a stable critical point of E γ but also remains as the global minimizer of the perturbed nonlocal problem (NLIP). Perhaps more interesting than its global minimality, analyzing the stability of the single cap we obtain a critical gamma value that we will denote by γ c where the single cap is stable for all γ < γ c and it is unstable for all γ > γ c . Hence, this value γ c serves as the borderline of stability of the single cap. We also establish instability of the double cap, i.e., the set whose boundary consists of two identical parallel circles, for sufficiently small and sufficiently large γ-values. Indeed, we will show that in the m = 0 regime the double cap is unstable for all γ > 0. Looking at (NLIP) on a curved surface clearly requires the inclusion of a curvature term in the second variation of E γ . However, the explicit knowledge of the Green's function for the Laplace-Beltrami operator on S 2 enables us to carry out the calculation of the second variation of E γ about the single cap and the double cap rather explicitly.
As in [30], for proving the global minimality of the single cap we use a "corralling" argument. After establishing the regularity of the free boundary of local minimizers we first show that if the set where a minimizer equals one is not uniformly close to that of the lamellar critical point, then necessarily, it has too much perimeter. Such a corralling step often occurs when working with sets of finite perimeter in the L 1 -topology, cf. e.g. [15,31], but here, the argument is in some ways more subtle due to the nonlocal term in the energy which prefers multi-component competitors. Next, we show that competitors that are uniformly close are in fact C 2 -close. Finally, exploiting the known stability of the lamellar critical point in the sense of second variation, we eliminate competitors that are C 2 -close. This scenario of converting stability to either local or global minimality in the context of (local) volume-constrained least area problems also arises in the works [11,18].
Our arguments for both global minimality and stability/instability of critical points require the regularity of minimizers of (NLIP). Given the well-developed regularity theory for isoperimetric domains, cf. e.g. [10,16], the issue here is to obtain a good estimate on the "excess-like" quantity that measures how far a set is from minimizing perimeter in a ball in terms of the radius of that ball, cf. (2.17). We show that even with the inclusion of the nonlocal term, it is still possible to obtain an estimate of order O(R 1+ ) for some > 0, hence allowing us to invoke the standard theory.
The paper is organized as follows: The next section is devoted to the regularity of minimizers of (NLIP). We state the formulas for the first and second variation of E γ in Section 3, and establish criteria for testing criticality and stability. In the fourth section we will investigate the single cap and demonstrate its stability and global minimality for any m ∈ (−1, 1) and for γ sufficiently small. Moreover we will show that the single cap critical point is unstable for large γ-values. In Section 5 we introduce the double cap in two configurations. After a comparison of total energies of these configurations depending on the (m, γ)-regime, we prove that in both configurations the double cap is unstable for small and large values of γ. Finally, in the sixth section we discuss some open problems.

Regularity
In this section, we establish the regularity of the set ∂{x ∈ S 2 : u(x) = 1} for any local minimizer u of (NLIP). For simplicity of presentation only let us give an equivalent reformulation of (NLIP) as a functional defined on sets of finite perimeter: We will phrase the regularity result in terms of L 1 -local minimizers, by which we mean any set Ω ⊂ S 2 of finite perimeter such that The main tool in the proof of regularity is the following lemma, cf. [9, Lemma 2.1], whose proof easily adapts to the case of a two-sphere. There exists two positive constants k 0 and l 0 , depending only on D and D ∩ L, such that for every k, |k| < k 0 , there exists a set F , with F = L outside D and Proof. From the definition of perimeter on a bounded subset we can conclude that there exists a vector field X ∈ X c (D) with X L ∞ (D) 1 such that Here X c (D) denotes the set of all compactly supported vector fields of class C ∞ on D.
Then we construct the corresponding flow deformation of L, whose boundary instantaneously moves according to the perturbation vector field X, by letting Ψ : for some τ > 0 and by defining L t := Ψ(L, t).
For t small, Ψ is a diffeomorphism. Hence, letting JΨ denote the Jacobian of Ψ in local coordinates, we have the following change of variables formulas where η ∈ L 1 (S 2 ; S 2 ) is obtained by differentiating the vector-valued measure ∇χ L with respect to the measure |∇χ L | (cf. [ has i-th component given by xj X (j) and K is defined by Moreoever, (DΨ) −1 = I − tH(x, t) for some H, where both |K| and |H| are bounded by a constant depending only on sup |∇X|, and therefore only on D and D ∩ L. With these, the proof of the lemma follows as shown in [9, Lemma 2.1].
We now state the regularity result for local minimizers on S 2 : Let Ω be an L 1 -local minimizer of (2.1). Then ∂Ω is of class C 3,α for some α ∈ (0, 1).

Proof.
Let Ω be an L 1 -local minimizer of E γ and let x 0 be any point of ∂Ω. Then let D ⊂⊂ U be such that x 0 ∈ D and D |∇χ Ω | > 0.
By Lemma 2.1, there exist two positive constants k 0 and l 0 , depending only on D and D ∩Ω, such that for every k, |k| < k 0 , there exists a set F , with F = Ω outside D and satisfying (2.3).
Fix R > 0 such that where ω 2 is the measure of the unit geodesic ball on S 2 and δ comes from (2.2). Moreover, letF minimize perimeter in B R (x 0 ) subject to the boundary values of Ω, i.e., we can use the same k 0 and l 0 as above withF replacing Ω in (2.3). Hence, for k := |Ω| − |F | ω 2 R 2 < k 0 , there exists a set G, with G =F outside D, and where the last condition follows from (2.7) with C 0 := 1 + l 0 D |∇χ Ω | ω 2 .
By (2.8), G is a competitor in (2.1) and we have that Thus, using the factF \ B R (x 0 ) = Ω \ B R (x 0 ) and G \ D =F \ D, along with (2.9), the inequality (2.11) becomes Hence, we get Now we estimate the nonlocal parts on the right-hand side of (2.12). To this end, let Hence for any p 1 we have (2.13) u Ω − u G L p (S 2 ) CR 2/p , through an appeal to (2.10). We take where κ is less than but as close as needed to 1 so that 1 < p < 2. Since H 1 imbeds continuously into L q for any q < ∞, using the Poincaré and Hölder inequalities we get that Thus, by combining (2.13)-(2.15) we obtain or in other words, since κ < 1, we have that for some > 0. Now, using (2.10), (2.16) and integration by parts, we obtain the following bound on the difference of the nonlocal parts: Returning to (2.12), this implies that Property (2.17) states that the boundary of the set Ω is almost area-minimizing in any ball. With this property in hand, the results of [14,32] apply, and we can conclude that ∂Ω is of class C 1,α . The C 3,α regularity of ∂Ω then follows from standard elliptic theory.

The first and second variations of (NLIP)
In this section, we will characterize the first and second variations of (NLIP). To fix notation, let us express the functional E γ as and N : where v : S 2 → R depends on u as the solution to the problem (1.2). First, let us note that we can write v in terms of the Green's function G = G(x, y) associated with the Poisson problem (1.2). Then, for each x ∈ S 2 , G(x, y) satisfies where δ x is a delta-mass measure supported at x. In particular, one can show, by writing out the Laplace-Beltrami operator in spherical coordinates explicitly, that for x, y ∈ S 2 where | · | denotes the Euclidean norm, that is, |x − y| is the chordal distance between x and y in R 3 . The functions G and v are then related by The first and second variations of P alone computed about a critical point of P have been investigated in [1] by calculating the first and second variations of area subject to fixed volume. However, for the first and second variations of E γ , one needs to proceed as the authors did in [7], that is, one needs to compute the second variation not about a critical point of P but rather about a critical point of E γ . To this end, in light of the reformulation given by (2.1), we view E γ as a functional depending on a set, say A ⊂ S 2 , through the formula Here A c denotes the complement of A, i.e., A c = S 2 \ A, and the 2-dimensional measure of A, H 2 (A), is compatible with the mass constraint on u, so that |A| = 2π(1 + m).
Given a set A ⊂ S 2 with C 2 boundary, we define an admissible perturbation of A as a family of sets {A t } t∈(−τ,τ ) for some τ > 0 so that the sets A t preserve measure to second order, i.e., H 2 (A t ) = H 2 (A) + o(t 2 ). In [7], to construct the admissible family {A t }, the authors use the ODE given by (2.4) to produce a flow deformation of A whose boundary instantaneously moves according to a perturbation vector field X, and which instantaneously preserves measure. They then apply a correction to insure it instantaneously preserves measure to second order as well. Indeed, in the proof of the following proposition, the perturbation vector field X = X f is chosen as X f = f ν on ∂A, for a given smooth function f satisfying the condition ∂A f (x)dH 1 x = 0, where ν denotes the outer normal to ∂A that is tangent to S 2 . Now, following the calculations in [7] and replacing the calculations for the local part with the calculations in [1], one easily obtains the following result.
Then for v solving (1.2) we have where λ is a constant and H denotes the geodesic curvature of ∂A. Moreoever, the second variaton of E γ about the critical point u is given by Here ∇ ∂A f denotes the gradient of f relative to the manifold ∂A, B ∂A denotes the second fundamental form of ∂A and ν denotes the unit tangent of S 2 which is normal to ∂A pointing out of A.
We finish this section with three remarks.
Remark. Here we want to note that if the critical point u of E γ is a local minimizer then Proposition 2.2 applies and we obtain the regularity of ∂A as stated in the hypothesis of the previous proposition.
Remark. In reference to the formula for the second variation above, a critical point Remark. We want to note that since ∂A is a curve on S 2 , the term B ∂A 2 in (3.7) involving the second fundamental form of ∂A coincides with the square of the geodesic curvature of ∂A, namely, B ∂A 2 = H 2 .

Single Cap: Stability and Global Minimality
In this section we will analyze the single cap, that is, the set whose boundary consists of a single circle. By rotational symmetries of the sphere we can assume that the boundary of the single cap is parallel to the equator, that is, using spherical coordinates we can assume that the boundary of the single spherical cap is the circle parametrized as (1, θ, φ 0 ) for θ ∈ [0, 2π] and for some fixed φ 0 ∈ (0, π). Let us fix some m ∈ (−1, 1) and define the function u S describing the single cap as follows: With this definition, the single cap then is the set Also note that, since 1 4π S 2 u dH 2 x = m, we get that φ 0 = arccos(−m). Before we start investigating the second variation of E γ about u S , we remark that since the function v S corresponding to u S through (1.2) clearly depends only on φ, one easily sees from (3.6) that u S is a critical point of E γ for all γ > 0. Now, writing out the Laplace operator in spherical coordinates, plugging into (1.2) and using the fact that v S is continuous at φ 0 and stays bounded as φ approaches 0 and π, we obtain an explicit formula for v S : Thus referring to (3.7), for any smooth function f satisfying ∂S f dH 1 x = 0, the second variation of E γ about u S takes the form Note that ∂S is a circle of radius √ 1 − m 2 ; hence expressing it in polar coordinates ( √ 1 − m 2 cos θ, √ 1 − m 2 sin θ) for θ ∈ [0, 2π] and invoking the identity Now, referring to (4.4), we can prove the following stability/instability result for the single cap S. Indeed this proposition provides a γ-value which defines the borderline for the stability of the single cap solution. Proof. First, let us note that for any θ = α we have the following identity (cf. page 190 in [34]): Also note that the singularity of the logarithm at θ = α is weak enough not to disturb integrability. Hence, in light of the orthonormality of {sin(nθ), cos(nθ)} n∈N , we get that where δ nm denotes the Kronecker delta.
As mentioned above, in [7] the second variation of E γ in Euclidean domains is calculated along a perturbation vector field X. Here, we take X = X f such that where ν denotes the outer normal to ∂S that is tangent to S 2 and 2π 0 f (θ) dθ = 0. Let us define g(θ) := f (θ) − (a 1 cos θ + b 1 sin θ), where a 1 and b 1 are the first Fourier coefficients in the expansion of f . Clearly, 2π 0 g(θ) dθ = 0. Let us also note that the perturbation of the boundary of the single cap by a vector field of the form (a 1 cos θ + b 1 sin θ)ν(θ) corresponds to a rotation (cf. [12,Section 4.2]). We will now argue that J γ (f ) > 0 for γ < γ c for some γ c > 0, provided g ≡ 0, that is, provided f does not correspond to a rotation.
Given any smooth function f satisfying 2π 0 f dθ = 0, let us look at its Fourier expansion given by Let Plugging f into (4.4) and using (4.6) then yields . For any γ < γ c , this implies that hence, u S is stable for γ < γ c .
To establish the instability of u S let us consider the function f 2 (θ) := sin 2θ on [0, 2π]. Clearly ∂S f 2 dH 1 x = 0. Plugging f 2 into (4.4) and invoking (4.6) yields Then, we get that J γ (f 2 ) < 0 for all γ > γ c , implying the instability of u S for γ sufficiently large.
We will now proceed to prove that for γ sufficiently small, the global minimizer u γ of E γ coincides with u S , the single cap.
We first note that we can immediately conclude that any sequence of minimizers {u γ } of (1.1) converges, after perhaps a rotation, to the single cap u S given by (4.1).

Proposition 4.2.
For any m ∈ (−1, 1), let {u γ } γ 0 be a sequence of minimizers of E γ . Then after perhaps a rotation, Proof. Since a uniform bound E γ (u γ ) < C is immediate in light of the minimality of u γ , one obtains a uniform BV -bound leading to convergence in L 1 of a subsequence as γ → 0. By the standard Γ-convergence argument, this limit must minimize P , the local isoperimetric problem, defined in (3.1). On the sphere, S 2 , however, it is known that the global minimizer of P is a single cap for any m ∈ (−1, 1), hence is equal to u S given by (4.1), after perhaps a rotation, cf. [29,Theorem 3.3].
With this in hand, one can also easily establish convergence of the functions v γ to v S as in [30, Proposition 3.2]. Proposition 4.3. For any minimizer u γ of E γ , there is a value α ∈ (0, 1) such that the corresponding solution v γ of (1.2) is bounded in C 1,α independent of γ.
Moreover, for a sequence of minimizers In particular, We now can state our result on the global minimality of the single cap.
Step 1. We first claim that there cannot exist a sequence of components S 1 γ ⊂ Ω γ whose area converges to zero as γ → 0.
To this end, we write Ω γ as a union of its connected components, i.e. Ω γ = Nγ j=1 S j γ . We first note that necessarily, N γ < ∞ since otherwise for fixed γ there would have to exist a sequence of components of Ω γ whose area (and perimeter) approach zero. This would be impossible in light of (3.6) and Proposition 4.3 which imply a (γ-dependent) bound on the L ∞ -norm of the curvature H γ of ∂Ω γ . Now we assume, by way of contradiction, that for a sequence of γ−values approaching zero, Ω γ has a component, say S 1 γ , with H 2 (S 1 γ ) → 0. Define S γ := Ω γ \ S 1 γ . Then χ Sγ → χ S in L 1 . Also, note that Per S 2 (S 1 γ ) → 0, for if not, that is, if say c := lim inf γ→0 Per S 2 (S 1 γ ) with c > 0, then since lim inf 1 2 Per S 2 (S γ ) This yields a contradiction to the fact that E γ (u γ ) → P (u S ) = 1 2 Per S 2 (S) = π √ 1 − m 2 by Γ-convergence. Now, the regularity of ∂Ω γ asserted in Proposition 2.2 and the fact that Per S 2 (S 1 γ ) → 0 imply that we can enclose S 1 γ in a geodesic disk whose radius approaches zero with γ. Shrink the disk until it touches ∂S 1 γ for the first time, and denote the radius of the shrunken disk by r γ and the point where the disk touches ∂S 1 γ by p γ . Then we have H γ (p γ ) 1 rγ and so by evaluating the criticality condition (3.6) at x = p γ , we see that λ γ → ∞ since v γ L ∞ is bounded independent of γ by Proposition 4.3. Returning to (3.6) for x = p γ , we conclude that in fact H γ (x) → ∞ for all x ∈ ∂Ω γ . Moreover, since H γ (p γ ) 1 rγ , for γ small enough, we have that, say, H γ (x) 1 4rγ for all x ∈ ∂Ω γ so S j γ is contained in a geodesic disk of radius 2r γ for each j ∈ {1, . . . , N γ }.
For a finer analysis, let ρ j γ := diam(S j γ ). Then S j γ is contained in a geodesic disk with radius ρ j γ . Now, define ρ γ := min{ρ j γ : j ∈ {1, . . . , N γ }} so that Then let the minimum ρ γ be attained at, say, j = j 0 , i.e., ρ γ is the distance between two points p j0 γ , q j0 γ on ∂S j0 γ . As S j0 γ is contained in a disk of radius ρ γ which must be tangent to ∂S j0 γ , say, at p j0 γ , we see that H γ (p j0 γ ) 1 ργ . Hence, using the L ∞ -bound on v γ and the criticality condition, we get that where C γ depends only on γ and v γ L ∞ , and C γ is O(γ). Thus at any point x ∈ ∂Ω γ we have and this gives that Thus for any j ∈ {1, . . . , N γ }, S j γ is contained in a geodesic disk of radius 2ρ γ . Using this fact we can find a lower bound on N γ depending on ρ γ as follows: Since 2π(1 + m) = H 2 (Ω γ ) = Nγ j=1 |S j γ | 2π(1 − cos 2ρ γ )N γ , we get that This lower bound on N γ with (4.8) will then imply that Hence Per S 2 (S γ ) → ∞ as γ → 0, which contradicts the fact that E γ (u γ ) → π √ 1 − m. Here we want to remark that the above argument also shows that there cannot be a sequence of components of the complement of Ω γ approaching zero in measure.
Step 2. We claim that Ω γ consists of precisely one component.
Suppose, for a contradiction, that Ω γ has at least two components. Without loss of generality, assume that Ω γ has two components, say Ω γ = S 1 ∪ S 2 . The proof generalizes easily to the case where Ω γ has three or more components. Note that, by the mass constraint 1 4π S 2 u dH 2 x = m, we have that Also, by the Bol-Fiala inequality, which is a generalization of the isoperimetric inequality (cf. [29]), for i = 1, 2, we have that Hence, we get that Taking the square root of both sides we then obtain But again, as in Step 1, this would result in lim inf γ→0 Per S 2 (Ω γ ) > 2π √ 1 − m 2 , a contradiction to the fact that E γ (u γ ) → π √ 1 − m 2 . Therefore Ω γ cannot have two or more components; hence, it consists of precisely one component.
Step 4. We next claim that the boundary of Ω γ is globally the graph of a function, that is, ∂Ω γ can be expressed as (1, θ, cot φ 0 + g γ (θ)) in spherical coordinates for some function Let Γ γ (s) = (θ γ (s), φ γ (s)) be a parametrization of ∂Ω γ by arc-length s. Let S 2 be oriented with the unit normal pointing outward and let σ γ denote the oriented angle ∠(∂ φ , dΓγ ds ), where ∂ φ is the unit tangent to S 2 given by (cos θ cos φ, sin θ cos φ, − sin φ) in spherical coordinates and dΓγ ds denotes the tangent vector to the curve Γ γ . Then, by [29, Proposition 1.1], we have (4.9) We will prove the claim of this step by considering two separate cases. First, let us consider the case where m ∈ (−1, 1) \ {0}. In this case, from Step 3, we have that H γ → cot φ 0 = 0. Suppose, for a contradiction, that σ γ (s γ ) = 0 for some s γ . Also assume, without loss of generality, that cot φ 0 > 0. The case cot φ 0 < 0 can be proved in the same way.
Therefore σ γ = 0. Similarly, one reaches a contradiction by assuming σ γ = π at a point. Thus, the tangent vector of Γ γ is never parallel to the vector ∂ φ and Γ γ can be expressed globally as the graph of a function depending on θ.
Therefore dσγ ds → 0 uniformly; and as σ γ (s γ ) = π 2 , we get that σ γ = 0 for γ small enough. Hence, as above, Γ γ can be expressed globally as the graph of a function of θ.
Step 5. Now we will show that for small γ > 0, the minimizers {u γ } of E γ are equal to u S , i.e., they represent the single spherical cap. For simplicity of presentation only, we will take m = 0 in the proof. Since we do not have any restrictions on the parameter regime of m, the proof for an arbitrary m ∈ (−1, 1) \ {0} follows after minor modifications.
Clearly, at t = 0 we have Taking the derivative one more time we get that for any t ∈ [0, 1] since cos π 2 + τ g γ (θ) g γ (θ) 0 for any θ ∈ [0, 2π]. Then, in particular at t = τ , we obtain the inequality Now using the definition of U (x, t), integrating by parts once and switching to spherical coordinates, we can rewrite the nonlocal part of the energy as Taking two derivatives with respect to t and evaluating at t = τ we get (4.12) Note that, by [ Returning to (4.12) then yields (4.13) where C 0 is a positive constant depending on V φ L ∞ , V L ∞ and g γ L 2 and hence is independent of γ or τ .
As in the proof of Proposition 4.1 we will write g γ (θ) = c 1 sin θ + c 2 cos θ +g γ (θ) for some functiong γ . We will assume, for a contradiction, thatg γ ≡ 0, that is, the perturbations S t of the single cap S do not correspond to rotations. Then (4.11) gives that where C 1 is a constant independent of γ and is positive by the Poincaré inequality. Now combining this with (4.13) and (4.10) yields For γ < C1 C0 , we then get a contradiction to the minimality of E γ (u γ ) = e γ (1) since g γ L 2 > 0; hence we conclude that necessarily,g γ ≡ 0, that is, u γ ≡ u S up to rotation.

Double Cap and Its Instability
Another interesting critical point of E γ is the double cap, that is, the set whose boundary consists of two parallel circles of same radius. The double cap appears in two configurations. In this section we will investigate the relation between those configurations and the instability of the double cap. As in the previous section, because of the rotational symmetries of the sphere, we can assume that the boundary components of the double cap are parallel to the equator, i.e., using spherical coordinates we can assume that the boundary components of the double spherical cap are the circles parametrized by (1, θ, φ 0 ) and (1, θ, π − φ 0 ) for θ ∈ [0, 2π] and for some fixed φ 0 ∈ (0, π/2).
Let us fix some m ∈ (−1, 1) and define the function u D,I describing the double cap of first configuration as follows: Similarly, the function u D,II describing the double cap of second configuration is defined as With these definitions, we introduce sets D I := {x ∈ S 2 : u D,I = 1} and D II := {x ∈ S 2 : u D,II = 1}. Even though it is the complement of the set D II which is of the form of a double cap on S 2 , with an abuse of language, we will call the set D II the double cap of type II for the sake of simplicity. Here we also want to note that, since 1 4π S 2 u dH 2 x = m, we have φ 0 = arccos 1−m 2 for type I, and φ 0 = arccos 1+m 2 for type II. Since the functions v D,I and v D,II corresponding to u D,I and u D,II , respectively, depend only on φ, using the facts that both v D,I and v D,II are continuous at φ 0 and π − φ 0 and stay bounded as φ approaches 0, we can solve (1.2) explicitly to obtain: While we will show instability of u D,I and u D,II for certain γ-regimes, it is still interesting to compare the total energies of two configurations of the double cap, namely, E γ (u D,I ) and E γ (u D,II ). To this end, adapting the notation in (3.1) and (3.2), let us write Clearly, for m = 0, both configurations have the same total energy. Also, after a simple calculation, we see that E γ (u D,I ) = E γ (u D,II ) along the curve defined in the (m, γ)−plane. Note that the curve γ(m) and the γ-axis divide the (m, γ)−plane into four regions: A, B, C and D (cf. Figure 1). By (5.5) and (5.6), we get that, in regions A and C, the second configuration has less energy than the first configuration, that is, E γ (u D,II ) < E γ (u D,I ), whereas in regions B and D we have E γ (u D,I ) < E γ (u D,II ). Now, we will establish the instability of u D,I and u D,II on certain regimes. By the fact that the boundary components of D I and D II are circles with radius sin φ 0 , referring to Let f be a smooth function defined on ∂D I such that ∂D I f dH 1 x = 0. Then, by the equations for B ∂D I 2 and (∇v D,I · ν)| ∂D I given above, (3.7) becomes (5.7) Using the equations for B ∂D II 2 and (∇v D,II · ν)| ∂D II this time, it is easy to note that J II,γ looks exactly the same as (5.7) except m is replaced by −m. Hence, the instability result of the next proposition for the double cap holds true for both the double cap of type I and II with a minor modification of switching m by −m.
Now we can state the instability result concerning the double cap. Proof. As noted above, we will prove this proposition only for u D,I . The proof for u D,II follows by switching m by −m. For the simplicity of presentation, let us define Let us also denote the boundary components of D I by Γ 1 and Γ 2 , that is, let ∂D I = Γ 1 ∪ Γ 2 .
Thus, when m = 0, u D,I is an unstable critical point of E γ for all values of γ > 0.

Closing Remarks
Being the first attempt to analyze (NLIP) on a curved surface, of course, our study leaves many unanswered questions that, we hope, will attract interest in the near future.
(i) From Proposition 4.1 we know that the single spherical cap is a stable critical point of (1.1) for any m ∈ (−1, 1) and for any γ < γ c . We also know that the single cap is the global minimizer of (NLIP) for sufficiently small γ-values. One can ask then whether the single cap remains as the global minimizer as γ approaches γ c . In other words, as we increase γ from 0 to γ c , when does the single cap lose its global minimality? Unfortunately, due to our use of an isoperimetric-type inequality in the proof of Theorem 4.4, at this moment, we are not able to quantify the γ-value which corresponds to the threshold of global minimality. (ii) One might also wonder about the nature of the bifurcation that occurs from the single cap at γ = γ c . Referring to the work [24] on wriggled lamellar patterns on the flat-torus, it seems reasonable to guess that the single cap might also bifurcate into a wriggled single cap as γ exceeds γ c . (iii) Proposition 5.1 gives a quite "negative" result regarding the double cap. Indeed, we show that for m = 0, the double cap is unstable for any γ > 0. At this point, it is unknown to us whether there is a regime of γ-values between γ 0 and γ 1 for which the double cap critical points are stable whenever m = 0. In [23] the authors show that striped patterns on a flat-torus possess stability as long as they have a certain number of interfaces. Also, in [2] the hedgehog defect structures (patterns where the diffuse interfaces are parallel annuli) were observed in self-consistent field theory simulations of lamellar block copolymers on a sphere. This suggests that (NLIP) might deliver stable critical points with parallel circles as the interfaces once the number of interfaces exceeds a fixed value.
Finally, we would like to remark that one needs to be careful when drawing an analogy between the lamellar patterns on the flat-torus and the hedgehog patterns on the sphere. One obvious difference is in the method of proving instability of the double cap when γ is small enough. Here, we have used the second variation evaluated at a function which takes on the values 1 and −1 on disconnected boundary components to obtain instability of the double cap for small γ-values. This idea does not work for double stripes on the flat-torus, as such functions correspond to translations and yield zero second variation. To our knowledge, instability of multiple striped patterns for small γ-values on the flat-torus is still an open problem. Another major difference is the dependence on the mass constraint. Our results here, especially the global minimality of the single cap, hold true for all values of m in the interval (−1, 1), whereas in the case of the flat-torus, the global minimality of the single stripe is obtained for a smaller interval of m-values (cf. [30,Theorem 3.3]). Indeed, the global minimizer of the local isoperimetric problem on the two-torus is either a disk or a stripe depending on the regime of m-values.