Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
CCOMPUTATION OF FREE BOUNDARY MINIMAL SURFACESVIA EXTREMAL STEKLOV EIGENVALUE PROBLEMS
CHIU-YEN KAO, BRAXTON OSTING, AND ´EDOUARD OUDET
Abstract.
Recently Fraser and Schoen showed that the solution of a certain extremal Stekloveigenvalue problem on a compact surface with boundary can be used to generate a free boundaryminimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meetsthe boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we developnumerical methods that use this connection to realize free boundary minimal surfaces. Namely, ona compact surface, Σ, with genus γ and b boundary components, we maximize σ j (Σ , g ) L ( ∂ Σ , g )over a class of smooth metrics, g , where σ j (Σ , g ) is the j -th nonzero Steklov eigenvalue and L ( ∂ Σ , g )is the length of ∂ Σ. Our numerical method involves (i) using conformal uniformization of multiplyconnected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solvinga boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developinggradient-based optimization methods for this non-smooth eigenvalue optimization problem. Forgenus γ = 0 and b = 2 , . . . , , , ,
20 boundary components, we numerically solve the extremalSteklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundaryminimal surface, which we display in striking images. For higher eigenvalues, numerical evidencesuggests that the maximizers are degenerate, but we compute local maximizers for the second andthird eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components. Introduction
Recently, A. Fraser and R. Schoen discovered a rather surprising connection between an extremalSteklov eigenvalue problem and the problem of generating free boundary minimal surfaces in theEuclidean ball [FS11, FS13, FS15]. These findings have been further developed [FTY14, FS19,GL20] and were recently reviewed in [Li19]. In this paper, we develop numerical methods tofurther investigate this connection. We first briefly review some of these previous results beforestating the contributions of the present work.
The extremal Steklov eigenvalue problem.
Let (Σ , g ) be a smooth, compact, connected Rie-mannian surface with nonempty boundary, ∂ Σ. The Steklov eigenproblem on (Σ , g ) is given by∆ v = 0 Σ(1a) ∂ ν v = σv ∂ Σ , (1b)where ∆ = | g | − ∂ i | g | g ij ∂ j is the Laplace-Beltrami operator and ∂ ν is the outward normal deriva-tive. The Steklov spectrum is discrete and we enumerate the eigenvalues, counting multiplicity, inincreasing order 0 = σ (Σ , g ) < σ (Σ , g ) ≤ σ (Σ , g ) ≤ · · · → ∞ . Date : July 31, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Steklov eigenvalue; eigenvalue optimization; free boundary minimal surface. a r X i v : . [ m a t h . SP ] J u l he Steklov spectrum coincides with the spectrum of the Dirichlet-to-Neumann operator Γ : H ( ∂ Σ) → H − ( ∂ Σ), given by the formula Γ w = ∂ ν ( H w ), where H w denotes the unique harmonic ex-tension of w ∈ H ( ∂ Σ) to Σ. The restriction of the Steklov eigenfunctions to the boundary, { v j | ∂ Σ } ∞ j =0 ⊂ C ∞ ( ∂ Σ), form a complete orthonormal basis of L ( ∂ Σ). A recent survey on Stekloveigenvalues can be found in [GP17].Here, for fixed surface Σ with genus γ and b boundary components, we consider the dependenceof the j -th Steklov eigenvalues on the metric, i.e. , the mapping g → σ j (Σ , g ). It is known that forany smooth Riemannian metric g , we have the following upper bound on the j -th Steklov eigenvaluein terms of the topological invariants γ and b ,(2) σ j (Σ , g ) L ( ∂ Σ , g ) ≤ π ( γ + b + j − ∀ j ∈ N . Here, L ( ∂ Σ , g ) is the length of ∂ Σ with respect to the metric g . This bound was proven by Weinstock[Wei54] for j = 1, γ = 0, and b = 1; by Fraser and Schoen [FS11] for j = 1 (see also [GP12]);and in generality by Karpukhin [Kar17]. It is then natural to pose the extremal Steklov eigenvalueproblem ,(3) ˜ σ (cid:63)j ( γ, b ) := sup g ˜ σ j (Σ , g ) , ˜ σ j (Σ , g ) := σ j (Σ , g ) L ( ∂ Σ , g ) , where g varies over the class of smooth Riemannian metrics on Σ. The existence of a smoothmaximizer in (3) was established in [FS15, Theorem 1.1] for oriented surfaces of genus 0 with b ≥ j = 1)eigenvalue. Free boundary minimal surfaces.
Denote the closed n -dimensional Euclidean unit ball by B n := { x ∈ R n : | x | ≤ } and the ( n − S n − = ∂ B n . Let M ⊂ B n bea d -dimensional submanifold with boundary ∂ M = M ∩ S n − . We say that M is a free boundaryminimal submanifold in the unit ball if(i) M has zero mean curvature and(ii) M meets S n − orthogonally along ∂ M .When d = 2, we call M a free boundary minimal surface in the unit ball or, more simply, a freeboundary minimal surface . For a good visual aid to understanding the definition of free boundaryminimal surfaces (and a peak at the results of this paper), we recommend the reader take a lookat the free boundary minimal surfaces displayed in Figures 13 and 14. Fraser and Schoen’s connection.
Fraser and Schoen observed that a d -dimensional submanifold M ⊂ B n with boundary ∂ M = M ∩ S n − is a free boundary minimal surface if and only if thecoordinate functions x i , i = 1 , . . . , n restricted to M are Steklov eigenfunctions with eigenvalue σ = 1. Furthermore, they showed the following theorem. Theorem 1.1 ([FS13]) . Let Σ be a compact surface with boundary. Suppose that g is a smoothmetric on Σ attaining the supremum in (3) for some j ∈ N . Let U be the n -dimensional eigenspacecorresponding to σ j (Σ , g ) . Then, there exist independent Steklov eigenfunctions u , . . . , u n ∈ U which give a (possibly branched) conformal immersion u = ( u , · · · , u n ) : Σ → B n such that u (Σ) isa free boundary minimal surface in B n and, up to rescaling of the metric, u is an isometry on ∂ Σ . Theorem 1.1 gives a method for using the solution of (3) to compute free boundary minimalsurfaces. The simplest such example is the equatorial disk, obtained as the intersection of B with any two-dimensional subspace of R . This can be constructed from Weinstock’s result thatinequality in (2) with j = 1, γ = 0, and b = 1 is attained only by the round disk, D [Wei54]. In this ase, for the eigenvalue ˜ σ (0 ,
1) = 2 π , we have the two-dimensional eigenspace given by span { x, y } .The equatorial disk is given as the map u : D → R , defined by u ( x, y ) = (cid:18) xy (cid:19) .For genus γ = 0 and b = 2 boundary components, the extremal metric is rotationally invariantand the corresponding free boundary minimal surface is the critical catenoid. We will discuss thisexample further in Section 3. For genus γ = 0 and b ≥ B and star-shaped with respect to the origin [FS13]. In [GL20], theauthors used homogenization methods to construct surfaces that have large first Steklov eigenvalue˜ σ . In particular, free boundary minimal surfaces of genus γ = 0 with particular symmetries ( e.g. ,symmetries of platonic solids) were constructed numerically. The authors proved that the firstnonzero Steklov eigenvalue, σ , of these surfaces is 1 and emphasized that it is not known whetherthese surfaces have extremal first eigenvalues among all surfaces with the same genus and numberof boundary components. We will compare our results to these surfaces in Section 5.In [FTY14], Fan, Tam, and Yu extended the study of (3) to higher values of j on the cylinder( γ = 0, b = 2) among rotationally symmetric conformal metrics. They obtained different resultsfor even and odd eigenvalues. They showed that the maximum of the ˜ σ j − , j ∈ N among allrotationally symmetric conformal metrics on the cylinder is achieved by the j -fold covering of thecritical catenoid immersed in R . The maximum of ˜ σ is not attained. The maximum of the ˜ σ j for j ≥ j -fold covering of the critical M¨obius band. These results will be further discussed in Section 3 andfurther compared to our computed surfaces in Section 5. Results and outline.
In this paper, we develop computational methods for solving the extremalSteklov eigenvalue problem (3) and thus generating free boundary minimal surfaces via Theorem 1.1.This approach is used to realize free boundary minimal surfaces beyond the known examples ofequatorial disks, the critical catenoid, the critical M¨obius band, and their higher coverings discussedabove.In Section 2, we explain how the conformal uniformization of multiply connected domains can beused to significantly reduce the complexity of the general Steklov eigenproblem (1) and extremalSteklov eigenproblem (3). The argument relies on two ingredients:(1) The uniformization result that for a smooth, compact, connected, genus-zero Riemanniansurface with b boundary components, (Σ , g ), there exists a conformal mapping f : (Σ , g ) → (Ω , ρI ), where Ω is a disk with b − ρI is a conformally flat metric.(2) The composition v ◦ f of a function v with a conformal map f is harmonic if and only if v is harmonic.Let D = { x ∈ R : | x | ≤ } be the unit disk andΩ c,r = D \ ∪ b − i =1 D i be a punctured unit disk with b − D i = D ( c i , r i ) = { x ∈ R : | x − c i | < r i } i = 1 , . . . , b − . This argument implies that it is sufficient to consider the family of (flat!) Steklov eigenproblems,∆ u = 0 Ω c,r (4a) ∂ n u = σρu ∂ Ω c,r , (4b) here ∆ is the Laplacian on Ω, ∂ n is the outward normal derivative, and ρ > γ = 0 is transformed to˜ σ (cid:63)j ( γ = 0 , b ) = max c i , r i , ρ ˜ σ j (5a) s.t. D i ⊂ D, i = 1 , . . . , b − D i ∩ D j = ∅ , i (cid:54) = j (5c) ρ ( x ) ≥ , x ∈ ∂ Ω c,r . (5d)Here, ˜ σ j = σ j L , σ j is the j -th nontrivial eigenvalue satisfying (4), and L = (cid:82) ∂ Ω c,r ρ ( x ) dx is thetotal length of ∂ Ω c,r . The first two constraints simply state that the holes are contained in thedomain and are pairwise disjoint.In Section 3, we explicitly solve the Steklov eigenvalue problem on a rotationally symmetricannulus ( i.e. , γ = 0, b = 2, c = 0, and ρ constant on each boundary component) and describe thecritical catenoid and its higher coverings in detail. These Steklov eigenvalues and correspondingfree boundary minimal surfaces will be used to verify our computational methods.In Section 4, we develop numerical methods for computing Steklov eigenvalues satisfying (4) onmultiply connected domains, computing the solution to the optimization problem (5), and the com-putation of free boundary minimal surfaces from the Steklov eigenfunctions. In brief, we use themethod of particular solutions to compute Steklov eigenvalues, gradient-based interior point meth-ods for the optimization problem, and compute the mapping to a surface by minimizing a particularenergy. These methods build on previous computational methods for extremal eigenvalue problemson Euclidean domains, including minimizing Laplace-Dirichlet eigenvalues over Euclidean domainsof fixed volume or perimeter [Oud04, Ost10, AF12, OK13, OK14, AO17, BBG17], maximizingSteklov eigenvalues over two-dimensional Euclidean domains of fixed volume [AKO17, BBG17].These methods have recently been extended to more general geometric settings. In particular,[KLO17] maximized Laplace-Beltrami eigenvalues over conformal classes of metrics with fixed vol-ume and compact Riemannian surfaces of fixed genus ( γ = 0, 1) and volume.In Section 5, we present the results of numerous computations. For genus γ = 0 and b =2 , . . . , , , ,
20 boundary components, we numerically solve the extremal Steklov problem (5) forthe first eigenvalue. We include figures displaying the optimal punctured disks and three linearly-independent eigenfunctions associated to the first eigenvalue, as well as tabulate the values ofthe obtained Steklov eigenvalues. We also plot the associated free boundary minimal surfaces,which are visually striking. Finally, in Section 5, we also present results for maximizing highereigenvalues. Here, numerical evidence suggests that the maximizers are degenerate, but we computelocal maximizers for the second and third eigenvalues with b = 2 boundary components and forthe third and fifth eigenvalues with b = 3 boundary components. For brevity, we were only ableto report the results for selected values of b and j ; the results of additional computations can befound on ´E. Oudet’s website [Oud20], along with gifs.We conclude in Section 6 with a discussion.2. The Euclidean Steklov eigenproblem
In Section 2.1, we explain how the conformal uniformization of multiply connected domainscan be used to significantly reduce the complexity of the general Steklov eigenproblem (1) andextremal Steklov eigenvalue problem (3) to obtain the Euclidean Steklov eigenproblem and (4)and extremal Steklov eigenvalue problem (5), respectively. In Section 2.2, we also compute theeigenvalue derivatives with respect to the density and shape parameters and discuss optimalityconditions for the extremal Steklov eigenvalue problem (5). .1. Conformal uniformization of multiply-connected surfaces and the Steklov eigen-problem.
The uniformization theorem for compact, genus-zero Riemann surface without bound-ary states that such surfaces can be conformally mapped to the Riemann sphere. Here, we use ageneralization of this result for multiply-connected surfaces; see [Hen86, Theorem 17.1b], [GL99],[ZYZ + Theorem 2.1 ([GL99]) . Suppose (Σ , g ) is a smooth, compact, connected, genus-zero Riemannsurface with b boundary components. Then Σ can be conformally mapped to a unit disk with b − circular holes. That is, there exists a punctured unit disk with b − holes, Ω c,r = D \ ∪ b − i =1 D i , anda conformal map f : (Σ , g ) → (Ω c,r , ρI ) , where ρI is a conformally flat metric. Furthermore, twosuch mappings differ by a M¨obius transformation.Remark . The uniqueness of the conformal map up to a M¨obius transformation means that itis possible to center one of the holes at the origin and center another hole on the positive x -axis.Thus, fixing these three parameters, the dimension of the parameter space of hole centers and radii { c i } b − i =1 ∪ { r i } b − i =1 , is 1 for b = 2 and 3 b − b ≥
3, which is the dimension of the conformalmodule.We now sketch a brief derivation of (4) from (1). Let f : (Σ , g ) → (Ω c,r , ρI ) be a conformalmapping. It is well-known that v = u ◦ f : Σ → R is harmonic if and only if u : Ω c,r → R is harmonic[Olv17]. This justifies (4a). We show (4b) on a flat domain for simplicity. Write x = f ( z ) and v ( z ) = u ( f ( z )) = u ( x ), so that ∇ z v ( z ) = Df ( z ) T ∇ x u ( f ( z )). Since Df ( z ) ν ( z ) = | Df ( z ) | n ( f ( z )),we have that σu ( f ( z )) = σv ( z )= ν T ( z ) ∇ z v ( z )= ν T ( z ) Df ( z ) T ∇ x v ( f ( z ))= | Df ( z ) | n T ( f ( z )) ∇ x u ( f ( z ))= | Df ( z ) | ∂ n u ( f ( z ))So, we obtain ∂ n u ( x ) = σρ ( x ) u ( x ), where ρ ( x ) = | Df (cid:0) f − ( x ) (cid:1) | − = | Dh ( x ) | , where h = f − .Remark 2.2 shows that our parameterization of Ω c,r is over-complete, as the following examplefurther demonstrates. Example . Denote Ω as an eccentric annulus with boundaries | z − c | < r and | z | < as an concentric annulus r < | x | < c , r , r are real numbers and x, z ∈ C . Aconformal mapping h : Ω → Ω is given by x = f ( z ) = z − a − az where a and r are determined by mapping c + r , c − r to r , − r and satisfy a = 1 + c − r − (cid:113)(cid:0) c − r (cid:1) − c c , and r = r + c − a − a ( r + c ) . In this example, z = h ( x ) = x + a ax , and ρ ( x ) = | z x | = (cid:12)(cid:12)(cid:12)(cid:12) − a (1 + ax ) (cid:12)(cid:12)(cid:12)(cid:12) . igure 1. A conformal mapping from an eccentric annulus to a concentric annulus.See Example 2.3.In Figure 1, the mapping is shown for c = r = and the resulting a = r = 2 − √ with boundary density ρ = 1 has the same Steklov spectrum as theconcentric annulus Ω with boundary density ρ ( x ) given above. In particular, this example showsthat the decomposition of perturbations of a metric into conformal and non-conformal directionsis not equivalent to either changing ( c , r ) or ρ , respectively. While changing ρ is a conformalperturbation, a change in ( c , r ) gives a perturbation to the metric that has components in boththe conformal and non-conformal directions.The following two examples illustrate what happens to the boundary density ρ when Σ becomes“pinched”. Example . We consider the conformal mapping h : D → Ω α from the unit disk | x | ≤ α , h ( x ) = 2 αx α + (1 − α ) x ;see [GLS16, AK19]. When α = 1, this is the identity mapping on the unit disk and as α → + , itmaps a unit disk to two “kissing” disks. In the left and center panels of Figure 2, the mapping isshown for α = . Here, we compute, ρ α ( x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:0) α − (1 − α ) x (cid:1) (1 + α + (1 − α ) x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let x = e iθ . In the right panel of Figure 2, we plot ρ α ( θ ) for α = , , and . We observe that ρ α ( θ ) becomes singular as α → + at θ = π and π .In Example 2.4, the density is singular at two points. The following example illustrates how thedensity function can become singular at a single point. Example . We consider a radius r := 0 .
833 disk, Ω = {| x | < r } , (see Figure 3(c)) and adomain Ω consisting of the union of two disks with radii r and 1 and a ‘neck’ of width 2 α (seeFigure 3(a)). Define the conformal mapping h : Ω → Ω as the composition of the two functions h = h ◦ h , where z := h ( y ) = y − icay + b + iαc, and y := h ( x ) = xr − i (1 − β )1 + i (1 − β ) xr . igure 2. A conformal mapping from a Hippopede shape to a unit disk. See Example 2.4.The constants a, b, c are chosen as a = 12 ( 1 α − r + 1 ) , b = 1 α − a, c = α + 4 aα − aα , so that h maps 1 , i, − , − i to α, i, − α, − r i , respectively. See Figure 3(a) and (b). The constant β is chosen so that Ω maps to a unit disk and the zero in Ω maps to − i (1 − β ) . See Figure 3(b)and (c).When β is small, this function maps points which are uniformly distributed on ∂ Ω topoints that accumulate near − i on the unit disk. The boundary density, ρ , can be obtained via theproduct rule, ρ ( x ) = | h x | = | h (cid:48) ( h ( x )) h (cid:48) ( x ) | , for | x | = r . As shown in Figures 3(d), the density reaches a large value at θ = π . Figure 3(e) shows the detailprofile of the density function about one.2.2. Eigenvalue derivatives with respect to the density and shape parameters.
In thissection, we consider σ and ˜ σ = σL as a function of ρ and the shape Ω c,r . We first compute thederivatives with respect to ρ . Proposition 2.6.
Let ( σ, u ) be a simple Steklov eigenpair, satisfying (4) , normalized so that (cid:82) ∂ Ω c,r ρu = 1 . Then the functionals ρ (cid:55)→ σ and ρ (cid:55)→ ˜ σ are Frech´et differentiable with deriva-tives (cid:104) δσδρ , δρ (cid:105) = − σ (cid:90) ∂ Ω c,r u ( x ) δρ ( x ) dx, (6a) (cid:104) δ ˜ σδρ , δρ (cid:105) = σ (cid:90) ∂ Ω c,r (cid:0) − Lu ( x ) (cid:1) δρ ( x ) dx. (6b) Proof.
We take variations of the formula σ = (cid:82) Ω c,r |∇ u | dx and use Green’s identity to obtain˙ σ = 2 (cid:90) Ω c,r ∇ u · ∇ ˙ u dx = − (cid:90) Ω c,r ˙ u ∆ u dx + 2 (cid:90) ∂ Ω c,r ˙ uu n dx = 2 σ (cid:90) ∂ Ω c,r ρu ˙ u dx. igure 3. A conformal mapping from a disk to a shape which is close to the unionof two disks. The choice of parameters are α = 0 . β = 0 .
1. See Example 2.5.From the normalization condition, (cid:82) ∂ Ω c,r ρu dx = 1, we obtain (cid:90) ∂ Ω c,r ˙ ρu dx = − (cid:90) ∂ Ω c,r ρu ˙ u dx, which gives the desired result. The derivative of ˜ σ is obtained via L = (cid:82) ∂ Ω c,r ρ dx and the productrule. (cid:3) We describe below optimality conditions when the multiplicity of the optimized eigenvalue isgreater than one. Our formulation is highly inspired by previous articles [ESI +
07, FS15, BO16].We first need the following regularity result; see [LP15, Theorem 3.2].
Lemma 2.7.
Let σ ( ρ ) be an eigenvalue of multiplicity p > of system (4) associated to a smoothdomain Ω with nonnegative boundary density ρ . Let δρ ∈ L ( ∂ Ω) and consider the eigenvaluesassociated to the densities ρ ε = ρ + εδρ for ε ∈ R . There exists ε > and nontrivial functions ( σ i ( ε )) ≤ i ≤ p and ( u i ( ε )) ≤ i ≤ p analytic on ( − ε , ε ) such that for all i = 1 , . . . , p : (a) σ i (0) = σ ( ρ ) , (b) The family { u ( ε ) , . . . , u p ( ε ) } is orthonormal in L ( ∂ Ω , ρ ε ) , (c) Every couple ( σ i ( ε ) , u i ( ε )) is solution of system (4) for the density ρ ε . We can now evaluate directional derivatives based on previous parametrizations: emma 2.8. Let σ be an eigenvalue of multiplicity p > of the weighted Steklov system (4) forsome nonnegative boundary density ρ . Denote by E σ the corresponding eigenspace. Let ρ ε = ρ + εδρ be a perturbation of ρ for some δρ ∈ L ( ∂ Ω) . Let ( σ i ( ε )) ≤ i ≤ p and ( u i ( ε )) ≤ i ≤ p be some smoothparametrizations as the ones given by Lemma 2.7. Then σ (cid:48) i = ddε σ i ( ε ) | ε =0 are the eigenvalues of thequadratic form q δρ defined on E σ ⊂ L ( ∂ Ω , ρ ) by q δρ ( u ) = − σ (cid:90) ∂ Ω u δρ dx. Moreover, the L ( ∂ Ω , ρ ) -orthonormal basis u (0) , ..., u p (0) diagonalizes q δρ on E σ .Proof. Let ( σ i ( ε )) ≤ i ≤ p and ( u i ( ε )) ≤ i ≤ p defined on ( − ε , ε ) for some ε > ε ∈ ( − ε , ε ), i = 1 , . . . , p and v ∈ L ( ∂ Ω , ρ ), we have from (4), that(7) (cid:90) Ω ∇ u i ( ε ) · ∇ v dx = σ i ( ε ) (cid:90) ∂ Ω u i ( ε ) vρ ε dx. Differentiationg this equality with respect to ε and evaluating at ε = 0 gives (cid:90) Ω ∇ u (cid:48) i (0) · ∇ v dx = σ (cid:90) ∂ Ω u i (0) vδρ dx + σ (cid:90) ∂ Ω u (cid:48) i (0) vρ dx + σ (cid:48) i (cid:90) ∂ Ω u i (0) vρ dx. Thus, with v = u j (0) and using (7) replacing i per j and v by u (cid:48) i (0), we obtain σ (cid:48) i (cid:90) ∂ Ω u i (0) u j (0) ρ dx = − σ (cid:90) ∂ Ω u i (0) u j (0) δρ dx which exactly proves that L ( ∂ Ω , ρ )-orthonormal basis u (0) , ..., u p (0) diagonalizes q δρ on E σ . More-over, the σ (cid:48) i are eigenvalues of this quadratic form. (cid:3) We can now establish optimality conditions with respect to the boundary density in case ofmultiple eigenvalues.
Proposition 2.9.
Let j ≥ and Ω a smooth domain of R . Assume a nonnegative ρ ∈ L ( ∂ Ω) maximizes the product σ j ( ρ ) L ( ρ ) among all nonnegative functions of L ( ∂ Ω) where L ( ρ ) = (cid:82) ∂ Ω ρ dx and σ j ( ρ ) is the j -th eigenvalues of system (4) . If σ j ( ρ ) is of multiplicity p > and E σ j itseigenspace, there exists a basis of p functions u , . . . , u p of E σ j which satisfy p (cid:88) i =1 u i ( x ) = 1 for all x ∈ ∂ Ω .Proof. The proposition is an almost direct consequence of Lemma 2.8 and of Hahn-Banach separa-tion theorem. Consider the convex hull K = Co (cid:8) u , u ∈ E σ j (cid:9) . We want to prove that the functionidentically equal to one belongs to K . If it is not the case, by Hahn-Banach theorem applied to thefinite dimensional normed vector subspace of C ( ∂ Ω) spanned by K and 1, there exists a function δρ ∈ C ( ∂ Ω) such that (cid:82) ∂ Ω δρ dx > u ∈ E σ j , (cid:90) ∂ Ω u δρ dx ≤ . This last inequality asserts that the quadratic form q δρ on E σ j has nonnegative eigenvalues. Thus,both the p eigenvalues and the weighted length increase in the direction of δρ . As a consequence,for ε small enough, the product of σ j ( ρ + εδρ ) L ( ρ + εδρ ) is strictly greater than σ j ( ρ ) L ( ρ ) due tothe strict inequality of the separation result which contradicts the optimality. (cid:3) o compute the derivatives of σ and ˜ σ with respect to the centers c and radii r , we first computethe shape derivative with respect to perturbations of the boundary of Ω c,r . This result extends aresult in [DKL14, AKO17, BBG17] to ρ (cid:54) = 1. Proposition 2.10.
Consider the perturbation x (cid:55)→ x + τ v . Then a simple (unit-normalized) Stekloveigenpair ( σ, u ) satisfies the perturbation formula (8) σ (cid:48) = (cid:90) ∂ Ω (cid:0) |∇ u | − ρ σ u − σκρu (cid:1) ( v · ˆ n ) + σρ t u ( v · ˆ t ) dx, where ˆ n is the outward unit normal vector, ˆ t denotes the tangential direction, and where κ is thesigned curvature of the boundary. We also have L (cid:48) = (cid:82) ∂ Ω κρ ( v · ˆ n ) − ρ t ( v · ˆ t ) dx .Proof. We follow the proof in [AKO17]. Let primes denote the shape derivative. From the identity σ = (cid:82) Ω |∇ u | dx , we compute σ (cid:48) = 2 (cid:90) Ω ∇ u · ∇ u (cid:48) dx + (cid:90) ∂ Ω |∇ u | ( v · ˆ n ) dx (shape derivative)(9a) = − (cid:90) Ω (∆ u ) u (cid:48) dx + 2 (cid:90) ∂ Ω u n u (cid:48) dx + (cid:90) ∂ Ω |∇ u | ( v · ˆ n ) dx (Green’s identity)(9b) = 2 σ (cid:90) ∂ Ω ρuu (cid:48) dx + (cid:90) ∂ Ω |∇ u | ( v · ˆ n ) dx (Equation (4)) . (9c)Differentiating the normalization equation, (cid:82) ∂ Ω ρu dx = 1, we have that2 (cid:90) ∂ Ω ρuu (cid:48) dx = − (cid:90) ∂ Ω ρ (cid:48) u + (cid:0) ∂ n ( ρu ) + κρu (cid:1) ( v · ˆ n ) dx, where κ is the curvature of the boundary and ρ (cid:48) = −∇ ρ · v . Extending ρ constantly in the normaldirection, we have ρ (cid:48) + ( v · ˆ n ) ρ n = − ρ t ( v · t ) where t denotes the tangential direction. We then havethat 2 (cid:90) ∂ Ω ρuu (cid:48) dx = (cid:90) ∂ Ω ρ t u ( v · t ) − (cid:0) ρuu n + κρu (cid:1) ( v · ˆ n ) dx. Combining this with (9), we obtain the desired result. (cid:3)
Using Proposition 2.10, we can now compute the derivatives of σ and ˜ σ for the domain Ω c,r = D \ ∪ b − i =1 D i with respect to a center c i and radius r i of D i as follows. To compute the derivativewith respect to r i , we choose a perturbation v so that v · ˆ n = − v · ˆ t = 0 on ∂D i . Then, noting that κ = − /r i , we obtain(10) ∂σ∂r i = − (cid:90) ∂D i |∇ u | − ρ σ u + σr i ρu dx. To compute the derivative with respect to c i , we take two perturbations v of the form v · ˆ n = cos θ and v · ˆ t = sin θ on ∂D i and v · ˆ n = sin θ and v · ˆ t = − cos θ on ∂D i , to obtain(11) ∇ c i σ = (cid:90) ∂ Ω (cid:18) |∇ u | − ρ σ u + σr i ρu (cid:19) (cid:18) cos θ sin θ (cid:19) + σρ t u (cid:18) sin θ − cos θ (cid:19) dx. emark . In [FS15], a detailed study of perturbations to the metric yield two conditions fora maximal Steklov eigenvalue. The first comes from the study of perturbations in “conformaldirections” and, as in Proposition 2.9, result in the existence of eigenfunctions { u j } nj =1 such thatthe map U = [ u | · · · | u n ] : Ω → B n satisfies U ( ∂ Ω) ⊂ S n − . The second condition comes fromthe study of non-conformal perturbations of the metric and give that the map U : Ω → B n hasisothermal coordinates, i.e. , satisfies | ∂ x U | = | ∂ y U | ,∂ x U · ∂ y U = 0 . Since a change in the parameters ( c, r ) gives a perturbation to the metric that has components inboth the conformal and non-conformal directions (see Remark 2.2 and Example 2.3), this secondcondition is nontrivial to obtain from (10) and (11).3.
Steklov eigenvalues of rotationally symmetric annuli and the critical catenoid
Here, we discuss the Steklov eigenvalues of rotationally symmetric annuli, the critical catenoid,and coverings of the critical catenoid. These results are also discussed in [FS11, FTY14] usingcylindrical coordinates, but it useful to review these computations and have them written in annularcoordinates for comparison and discussion; see also [Mar14, Dit04].3.1.
Steklov eigenvalues of rotationally symmetric annuli.
Here, for s ∈ (0 , A s = { ( r, θ ) : r ∈ [ s, } , and explicitly compute Steklov eigenvalues satisfying[ r − ∂ r r∂ r + r − ∂ θ ] u = 0 ( r, θ ) ∈ A s , (12a) ∂ ν u = σρ s u r = s, (12b) ∂ ν u = σρ u r = 1 . (12c)Note that if ( σ, u ) is an eigenpair satisfying (12) with parameters ( s, ρ s , ρ ), then for α > σ/α, u ) is an eigenpair satisfying (12) with parameters ( s, αρ s , αρ ). Using separation of variables,we obtain general solutions to the Laplace equation of the form u ( r, θ ) = C + C log( r ) + ∞ (cid:88) k =1 ( C r k + C r − k )( C cos kθ + C sin kθ ) , where C , . . . , C are constants. Using the Steklov boundary conditions, we can determine theeigenpairs, ( σ, u ). Of course, there is a trivial eigenvalue, σ = 0 with corresponding constanteigenfunction. There is another eigenpair with eigenfunction that is constant in θ , given by σ = ρ + sρ s ρ ρ s s s − , u ( r, θ ) = 1 + σρ log r. We note that L = 2 π ( ρ + sρ s ), so that˜ σ = σL = 2 π ( ρ + sρ s ) ρ ρ s s s − . For each k = 1 , , . . . , there are also eigenfunctions that are oscillatory in θ of the form u ( r, θ ) = ( Ar k + Br − k ) { cos kθ, sin kθ } , igure 4. (left) Length normalized Steklov eigenvalues of the annulus, A s forvarying inner radius s . The blue lines represent multiplicity two eigenvalues fordifferent values of k , while the red line represents a multiplicity one eigenvalue. (right) For s = 0 . A , B are constants. Here, the brackets indicate that we can choose either cos or sin; thecorresponding eigenvalue has multiplicity two. Using the boundary conditions we obtain the 2 × (cid:18) k − k − ks k − ks − k − (cid:19) (cid:18) AB (cid:19) = σ (cid:18) ρ ρ ρ s s k ρ s s − k (cid:19) (cid:18) AB (cid:19) . This is equivalent to the eigenproblem kρ sρ s sinh( − k log s ) (cid:18) sρ s s − k + ρ s k − sρ s s − k − ρ s − k − sρ s s k − ρ s k sρ s s k + ρ s − k (cid:19) (cid:18) AB (cid:19) = σ (cid:18) AB (cid:19) , from which one obtains the real positive eigenvalues σ k, ± = k ρ sρ s coth( − k log s ) (cid:20) ρ + sρ s ± (cid:113) ( ρ + sρ s ) − ρ sρ s tanh ( − k log s ) (cid:21) . In Figure 4(left), for ρ s /ρ = 11 . s . The eigenvalue corresponding to the radially symmetric eigenfunction isplotted in red. The thin vertical line indicates the value s = 0 . s , thefirst Steklov eigenvalue has multiplicity three and length-normalized eigenvalue ˜ σ = 10 . π .3.2. Extremal eigenvalues for rotationally symmetric annuli.
We consider the extremaleigenvalue problem for rotationally symmetric annuli,max s,ρ s ,ρ ˜ σ j , ˜ σ j := σ j L. (13) ere, σ j is assumed to satisfy (12).3.2.1. The first eigenvalue.
We first consider j = 1. By the symmetry of ρ and sρ s , we obtain theoptimality condition sρ s = ρ =: ρ. In this case, we have the two length-normalized eigenvalues and associated L ( ∂ Ω , ρ )-normalizedeigenfunctions σ , − L = 4 π − s s , u ( r, θ ) = 1 √ πρ cosh (cid:16) log r √ s (cid:17) cosh (log √ s ) { cos θ, sin θ } σL = 8 π log s − , u ( r, θ ) = 1 √ πρ log r √ s log √ s . The two values of σL are equal when s is the unique solution of the transcendental equation1 + s − s = − log √ s, s > . The solution is approximately given by s = 0 . U : A s → B , defined by U ( r, θ ) = cosh (cid:16) log r √ s (cid:17) √ cosh (log √ s )+log √ s cos θ cosh (cid:16) log r √ s (cid:17) √ cosh (log √ s )+log √ s sin θ log r √ s √ cosh (log √ s )+log ( √ s ) , ( r, θ ) ∈ A s . Note that this map has coordinates that are linear combinations of the above eigenfunctions. Onecan check that these are isothermal coordinates, i.e. , | ∂ r U ( r, θ ) | = r − | ∂ θ U ( r, θ ) | , ∀ ( r, θ ) ∈ A s ,∂ r U ( r, θ ) · r − ∂ θ U ( r, θ ) = 0 , ∀ ( r, θ ) ∈ A s , and satisfy U ( ∂A s ) ⊂ S ⊂ R , i.e. , | U (1 , θ ) | = | U ( s, θ ) | = 1 , ∀ θ ∈ [0 , π ] . Furthermore, it is not difficult to check that U ( A s ) is the critical catenoid. That is, U ( A s ) = C α ∗ where C α = (cid:26) x ∈ R : (cid:113) x + x = α cosh (cid:16) x α (cid:17)(cid:27) , α > , is a catenoid and the critical catenoid is the catenoid with α = α ∗ = (cid:0) β + cosh β (cid:1) − where β = − log √ s ≈ . β = coth β . It is known that the critical catenoidis a free boundary minimal surface [FS15]. σ j L s ρ s /ρ multiplicity1 10 . . . π ∞
33 20 . . . . . . . . . . . . Figure 5. (top)
The value of ˜ σ j = σ j L for s ∈ [0 . , .
9] and ρ s ρ ∈ [1 ,
15] for j = 1 , . . .
6. The black dots indicate the maximum values in the domain that isshown. (bottom)
A table with the maximum values of ˜ σ j , the values of s and ρ s ρ attaining the maximum, and the multiplicity of the eigenvalue at the maximum. SeeSection 3.2.2.3.2.2. Higher eigenvalues.
For larger values of j , we numerically solve (13). In Figure 5, we plotthe value of ˜ σ j as a function of s and ρ s /ρ for j = 1 , . . . ,
6. The maximum value of σ j L is indicatedand data for the maximum values is also tabulated. Observe that for j = 1 , , . . . ,
6, we have that sρ s = ρ .For odd j = 2 m − m ∈ N , from the results of Fan, Tam, and Yu [FTY14], we have thatthe extremum is attained at the crossings of the two length-normalized eigenvalues with associated L ( ∂ Ω , ρ )-normalized eigenfunctions σ j, − L = 4 πj − s j s j , u ( r, θ ) = 1 √ πρ cosh (cid:16) j log r √ s (cid:17) cosh ( j log √ s ) { cos jθ, sin jθ } σL = 8 π log s − , u ( r, θ ) = 1 √ πρ log r √ s log √ s . The two values of σL are equal when s is the unique solution of the transcendental equation1 + s j − s j = − log s j , s > . e obtain ˜ σ m − = m ˜ σ (cid:63) , for m ≥
1. The extremal metric is achieved by the m -fold cover of thecritical catenoid.For even j , Fan, Tam, and Yu [FTY14] show the following. For j = 2, the extremal value is notattained among rotationally symmetric annuli and for even j ≥
4, the extremal value is attained.For m ≥
2, we have σ m L = 4 mπ tanh( mT m, (1)2 ), where T m, (1) is the unique positive root of m tanh ms tanh s = 1 The extremal metric is achieved by the critical m -M¨obius band, which havegenus γ = 1. These are not in the class of surfaces relevant to our later computational examples.4. Computational Methods
In Section 2, we described how conformal maps could be used to reduce the general Stekloveigenproblem (1) to the Euclidean Steklov eigenproblem (4). In this section, we describe thecomputational methods used to solve the Euclidean Steklov eigenproblem (4), optimization methodsused to solve the extremal eigenvalue problem (3), and methods for computing the minimal surfacefrom the Steklov eigenfunctions.4.1.
Solving the Euclidean Steklov eigenproblem (4) . We use the method of particular solu-tions to solve the Steklov eigenproblem (4). This method for multiply-connected Laplace problemswas recently discussed in [Tre18]. The methods rely on the following Theorem.
Theorem 4.1 (Logarithmic Conjugation Theorem [Tre18]) . Suppose Ω is a finitely connectedregion, with K , . . . , K N denoting the bounded components of the complement of Ω . For each j ,let a j be a point in K j . If u is a real valued harmonic function on Ω , then there exist an analyticfunction f on Ω and real numbers c , . . . , c N such that u ( z ) = Re f ( z ) + c log | z − a | + · · · + c N log | z − a N | , ∀ z ∈ Ω . Let M ∈ N ∗ and consider some fixed punctured disk Ω c,r . Based on Theorem 4.1, we definethe finite basis B to approximate solutions of eigenvalue problem (4) as the union of the harmonicrescaled real and imaginary parts of the functions(14) B = M (cid:91) j =0 (cid:8) z (cid:55)→ z j (cid:9) k − (cid:91) i =1 M (cid:91) j =1 (cid:26) z (cid:55)→ z − c i ) j (cid:27) k − (cid:91) i =1 { z (cid:55)→ log | z − c i |} . For instance, we rescaled the basis polynomial Re (cid:16) z − c ) (cid:17) by a factor r so that this basis functiontakes values of order 1 on the second circle. Consider now ( p l ) ≤ l ≤ L a uniform sampling with respectto arc length of ∂ Ω c,r . Using B , we approximate solutions of eigenvalue problem (4b) by the solutionof the non symmetric square generalized eigenvalue problem(15) B T A u d = σ d B T B u d , where A = (cid:16) ∂φ∂n ( p l ) (cid:17) ≤ l ≤ L, φ ∈B and B = ( φ ( p l )) ≤ l ≤ L, φ ∈B . Example . To illustrate the complexity of the approach to obtain a fine approximation of eigen-values, we considered a circular domain with four holes and L = 5000 points; see Figure 6(left).We evaluated the first six nontrivial eigenvalues with a high number of B elements for M = 50. InFigure 6(right), you can observe the evolution of the error with respect to M for M taking valuesfrom 2 to 10. Taking the converged values as an approximation of the exact ones, in this specificexample, it can be observed that with M = 10 the error is already smaller than 10 − . Here, thefirst nontrivial eigenvalue has multiplicity two, so the curves are almost indistinguishable. igure 6. An illustration of the convergence of the eigenvalues with respect to thenumber of basis functions for a non-simply connected domain. See Example 4.2. j α = 0 . α = 0 . α = 0 . α = 01 0.37968380 0.32288183 0.28797139 02 1.99258587 1.99688224 1.99338590 23 2.02351398 2.00917719 1.99906424 24 2.20444005 2.66795651 2.09627138 25 2.78126086 2.66795651 2.60980134 26 3.99885096 3.99479457 3.98132439 47 4.09199872 4.03602674 4.00214005 48 4.36831843 4.24271684 4.18039135 49 4.95936215 4.80367369 4.69676874 410 6.02510373 6.00554908 6.01439273 6 Table 1.
The first ten nontrivial Steklov Eigenvalues, σ j , of the Hippopede domain,Ω α , for α = 0 .
1, 0 .
06, 0 .
04. The last column are the values, known analytically, thatappear in the limit as α → Example . We now consider a geometric convergence study related to Example 2.4; see alsoFigure 2. Using the mapping from the unit disk to the Hippopede domain, Ω α , we study the limitas α →
0. Our computations are performed on the unit disk with non-constant density, ρ , asgiven in Example 2.4. In the limit, the density becomes singular, and the purpose of this exampleis to illustrate that a weakness of our numerical method is that we cannot accurately computeeigenvalues of pinched domains ( α →
0) or, equivalently, if the density is singular. The results aredisplayed in Table 4.1. The values for the disjoint union of two radius 0.5 disks, obtained in thelimit α →
0, are given in the rightmost column of Table 4.1. We note a very slow convergence ofthe eigenvalues as α → Optimization methods for extremal Steklov eigenvalues (5) . We used gradient-basedoptimization methods to solve the extremal Steklov eigenvalue problem (5). We first describe ourparameterization of the boundary4.2.1.
Parameterizing the geometry.
Let ρ ∈ L ∞ ( ∂ Ω c,r ) be the boundary density and denote therestriction of ρ to the i -th disk boundary by ρ i = ρ | ∂D ( c i ,r i ) , i = 1 , . . . , k − . inally, denote D k := D and ρ k the restriction of ρ to ∂D k . Thus, if Ω c,r has b boundary compo-nents, the geometry is described by the parameters { c i } b − i =1 , { r i } b − i =1 , and { ρ i ( x ) } bi =1 . Since ∂D ( c i , r i ) ∼ = S , we expand each ρ i in the truncated Fourier series ρ i ( θ ) = A i, + N (cid:88) (cid:96) =0 A i,(cid:96) cos( (cid:96)θ ) + B i,(cid:96) sin( (cid:96)θ ) , θ ∈ [0 , π ] . From Remark 2.2, it would be possible to center one of the holes at the origin and another onthe positive x -axis. However, we found that the representation of the boundary density ρ for finitebasis size (finite N ) was better without fixing these centers.4.2.2. Gradient based optimization methods.
As in [AKO17], to handle multiple eigenvalues, wetrivially transform (5) into the following problemmax t (16a) s.t. t ≤ σ i L i = j, j + 1 , . . . , (16b)We approximated the positivity constraint ρ ≥ L sample points,(16c) ρ ( p (cid:96) ) (cid:96) = 1 , . . . , L. This approximation leads to linear inequalities with respect to the coefficients ( A i,l , B i,l ) only. Wealso augment the previous optimization problem with the geometrical constraints in (5) by imposingthe (few) quadratic constraints on the variables ( c i , r i ) ≤ i ≤ k − : | c i | < (1 − r i ) i = 1 , . . . , k − , (16d) | c i − c j | > ( r i + r j ) i, j = 1 , . . . , k − , j (cid:54) = i. (16e)Using the derivatives computed in (6), (10), and (11), together with the interior point methodimplemented in [BNW06], we solved (16). All results of section 5, have been obtained with thefollowing parameters: M = 30 (maximal order of basis elements), L = 10 (number of samplingpoints) and at most 5 ,
000 iterations to reach a first order optimality condition criteria to a relativeprecision of 10 − . Observe that in all cases, we were able to recover the multiplicity three of theoptimal eigenvalue up to 6 digits.In our implementation, the computational cost is proportional to the number of connected com-ponents of the boundary. For instance, one hour of computation on a standard laptop was requiredto obtain the desired precision for three boundary components.4.3. Computing the free boundary minimal surface from the Steklov eigenfunctions.
At this point we assume that we have successfully solved the extremal Steklov problem (5) and wantto use Theorem 1.1 to compute the associated free boundary minimal surface using the Stekloveigenfunctions.Let σ denote the optimal eigenvalue and assume that it has multiplicity n . Define the mapping v = [ v , . . . , v n ] : Ω → R n , where { v i } ni =1 is some choice of basis for the n -dimensional eigenspace.For A ∈ R n , we consider the map u A : Ω → R n , defined by u A ( x ) = [ v ( x ) , . . . , v n ( x )] A, x ∈ Ω . We want to identify the matrix A so that the map u A = u = [ u , . . . , u n ] satisfies the spherical andthe isothermal coordinate conditions, | ∂ r u ( r, θ ) | = r − | ∂ θ u ( r, θ ) | , ∀ ( r, θ ) ∈ Ω r,c (17a) ∂ r u ( r, θ ) · r − ∂ θ u ( r, θ ) = 0 , ∀ ( r, θ ) ∈ Ω r,c . (17b) igure 7. Optimal disks configurations for 2 to 9 and 12 (last bottom right picture)connected components of the boundary. The red cross indicates the center of theunit disk.To identify the matrix A , so that u A : Ω → R n satisfies (17), we construct the objective function(18) J ( A ) = (cid:90) ∂ Ω W ( u A ( x )) dx + (cid:90) Ω (cid:0) | ∂ r u A ( r, θ ) | − r − | ∂ θ u A ( r, θ ) | (cid:1) + | ∂ r u A ( r, θ ) · r − ∂ θ u A ( r, θ ) | dx, where W ( u ) = ( | u | − . We then minimize J ( A ) over A ∈ R n × n . In all experiments in section5, using this selection process, we were able to obtain three eigenfunctions which take values inthe sphere on ∂ Ω to an absolute pointwise error bounded by 10 − . Moreover, since we have aparameterization of the surface, using the well-known analytic formula, we were able to computethe mean curvature of the surfaces, which in all cases was bounded by 10 − . The mean curvature andthe Gaussian curvature are plotted on the free boundary minimal surface at [Oud20]. Additionally,the angle that the boundary makes with the normal vector to the sphere is less than one degree. igure 8. Three linearly independent eigenfunctions associated to the first eigen-value for two and three boundary components.
Figure 9.
Three linearly independent eigenfunctions associated to the first eigen-value for four and five boundary components. igure 10. Optimal densities for two and three boundary components.
Figure 11.
Optimal densities for four boundary components. igure 12. Optimal densities for five boundary components.
Figure 13.
Approximation of a minimal surface in the ball with three and fourconnected components of the boundary. igure 14. Approximation of a minimal surface in the ball with five (first row),twelve (second row, two first views) and fifteen connected components of the bound-ary. b ˜ σ compare to [GL20] BC center configuration2 10 . . . . π ≈ . . . . π ≈ . . . . π ≈ . ∗ square antiprism (not regular)9 16 . . . π ≈ . . . . π ≈ . ∗ irregular, not dodecahedron Table 2.
For different number of boundary components b , we report the value ofthe first nontrivial normalized Steklov eigenvalue ˜ σ = σ L , the value obtained by[GL20], and the configuration of the centers of the boundary components. For b = 8and b = 20, our configuration of boundary components differs from [GL20], so thevalues should not be directly compared (indicated with an asterisk). igure 15. Convex polytopes associated to the center of mass of boundary con-nected components of minimal surfaces in the ball.
First row.
Four (first plot)and six boundary connected components (the two remaining plots).
Second row.
Two views of a square antiprism associated to a minimal surface with a boundarymade of height connected components and an icosahedron associated to a minimalsurface with twelve connected components in its boundary (last plot).5.
Numerical solutions of the extremal Steklov eigenvalue problem and thecorresponding free boundary minimal surfaces
In this section, we describe the solutions for the extremal Steklov eigenvalue problem (5), forvarious number of boundary components (BC), b , and eigenvalue number, j , and the correspondingfree boundary minimal surfaces (FBMS).5.1. First nontrivial eigenvalue ( k = 1 ). We first consider the first nontrivial eigenvalue ( k = 1)for varying numbers of BC, b = 2 , . . . , , , ,
20. In each case, the multiplicity of the extremaleigenvalue is three, as expected [FS15]. In Figure 7, we plot the optimal punctured disks, Ω c,r , for b = 2 , . . . , b = 12 BC. In Figures 8 and 9, we plot three linearly independent eigenfunctionsassociated to the first eigenvalue on their respective punctured disk for b = 2 , , , b , the corresponding optimal densities are plotted in Figures 10, 11, and 12. In Figures 13and 14, we plot the corresponding (approximate) FBMS in the ball for b = 3 , , , ,
15 BC. Inall cases, the BC of the FBMS are positioned at very symmetric locations, as further illustratedin Figure 15. Values of ˜ σ and additional information about these configurations are recorded inTable 4.3. Additional figures, including gifs, can be found at [Oud20] and were not included herefor brevity. igure 16. Six linearly independent eigenfunctions associated to the third eigen-value for three boundary components.We now make a few more detailed remarks for the problem with the various number of BC, b ,considered, especially for values of b that are related to the platonic solids. For some values of b ,we also compare to the FBMS discussed in [GL20].For b = 2, we recover the critical catenoid, the known FBMS [FS15] that we also discussed inSection 3. Note that in Figure 7 the hole is centered within the disk and in Figure 10, the density isconstant on each BC. The eigenfunctions plotted in Figure 8 exhibit symmetries and are explicitlygiven in Section 3; see Figure 4(right).For b = 3, the FBMS has BC positioned with centers on an equilateral triangle inscribed on agreat circle of the sphere; see Figure 13. Interestingly, the holes in the domain, Ω c,r , are slightlyasymmetrically configured; see Figure 7. The densities plotted in Figure 10 do not exhibit symme-try. The eigenfunctions plotted in Figure 8 do not exhibit symmetries, but this could be a resultof our (arbitrary) choice within the three dimensional eigenspace.For b = 4, the FBMS has BC positioned with centers at the vertices of a regular tetrahedron;see Figure 13. This is further illustrated in Figure 15, where the BC are overlaid on a regulartetrahedron. A similar minimal surface was computed in [GL20] and the value of ˜ σ is within 10 − ;see Table 4.3. In Figure 7, the holes in the domain, Ω c,r , are slightly asymmetrically configured.In Figure 11, the density on the outer boundary is nearly constant and the densities on the innerboundaries are similar to each other. There is no clear structure to the eigenfunctions potted inFigure 9.For b = 5, the FBMS has BC positioned with centers at the vertices of a triangular bipyramid;see Figure 14. In Figure 7, the holes in the domain, Ω c,r , are not only asymmetrically configured,but the radii of the holes vary. In Figure 11, the density on the outer boundary is nearly constantand the densities on the inner boundaries are similar to each other. Again, the eigenfunctionsplotted in Figure 9 do not appear to be structured.For b = 6, the FBMS has BC positioned with centers at the vertices of a regular octahedron;see Figure 14. This is further illustrated in Figure 15, where the BC are overlaid on a regular igure 17. Nine first linearly independent eigenfunctions associated to the fiftheigenvalue for three boundary components.octahedron. Again, a similar minimal surface was computed in [GL20] and the value of ˜ σ is within5 × − ; see Table 4.3. In Figure 7, the holes in the domain, Ω c,r , are slightly asymmetricallyconfigured; there is a small hole near the origin and four holes of equal radii roughly centered atthe vertices of a square. In Figure 11, the density on the outer boundary is nearly constant andthe densities on the inner boundaries are similar to each other.For b = 7, the FBMS has BC positioned at the vertices of a pentagonal bipyramid. Figures ofthe FBMS can be found at [Oud20]. In Figure 7, the domain, Ω c,r , has a small (uncentered) holesurrounded by five holes.For b = 8, the FBMS has BC positioned at the vertices of a square antiprism; see [Oud20]and Figure 15. Interestingly, we obtain ˜ σ ≈ . σ ≈ . c,r , has three smaller holes surrounded byfour larger holes. igure 18. Two distinct approximations of a minimal surface in the ball with threeconnected components of the boundary associated to the third and fifth Stekloveigenvalues.For b = 9, the FBMS has BC positioned at the vertices of a triaugmented triangular prism.Figures of the FBMS can be found at [Oud20]. In Figure 7, the domain, Ω c,r , has three smallerholes surrounded by five larger holes.For b = 12, the FBMS has BC positioned at the vertices of a regular icosahedron; see Figure 14.This is further illustrated in Figure 15, where the BC are overlaid with a regular icosahedron. Asimilar minimal surface was computed in [GL20] and the value of ˜ σ is within 10 − ; see Table 4.3.In Figure 7, the domain, Ω c,r , have one small uncentered hole, surrounded by five medium-sizedholes, surrounded by five larger holes.For b = 15 the FBMS is plotted in Figure 14. The FBMS has BC that are positioned withcenters with triangular symmetry.For b = 20 the FBMS has irregularly located BC; a figure can be found at [Oud20]. Interestingly,we obtain ˜ σ ≈ . σ ≈ . b = 8 and 20 do not have BCs centered at the vertices of aplatonic solid. It seems that the positions of the BCs are related to the minimizing configurationsfor Thompson’s problem; known as the Fekete points [Fek23, Bro20].We note that the FBMS obtained here are closely related to the k -noid surfaces; see [Web20]. Itmay be appropriate to the FBMS computed here as critical k -noids . .2. Higher eigenvalues ( j ≥ ). Here, we consider the extremal Steklov eigenvalue problem (5),for higher eigenvalues, ˜ σ j , j ≥
2. Less in known in this case and, in particular, the multiplicity ofthe optimal eigenvalue, and hence the dimension in which the FBMS exists, is unknown.We recall from [FTY14] (see also Section 3) that by maximizing σ j for odd j among rotationallysymmetric annuli yields an j +12 covering of the critical catenoid, a FBMS with b = 2 boundarycomponents and j -th normalized Steklov eigenvalue,˜ σ j = j + 12 ˜ σ (cid:63) , j odd . We also recall the result of [FS19, Theorem 5.3], that the degenerate surface consisting of the criticalcatenoid glued to j − b = 2 boundary components in 3 + 2( j − j -th normalized Steklov eigenvalue,˜ σ j = ˜ σ + ( j − π. We first consider b = 2 BC and eigenvalue j = 2. In this case, the density ρ on the outerboundary of the punctured disk becomes degenerate and resembles the ρ discussed in Example 2.5and displayed in Figure 3. We believe that this ρ corresponds to the critical catenoid glued to adisc, but this is difficult to resolve using our numerical method; see Example 4.3. For other highereigenvalues, we see similar phenomena for some initializations of ρ . However, there are a few valuesof eigenvalue number j and BC b , that give interesting local maximizers and are very robust withrespect to the initialization.For b = 2 BC and j = 3 eigenvalue, we obtain a double covering of the critical catenoid asobtained by [FTY14]; see [Oud20]. The value obtained is ˜ σ = 2˜ σ ∗ ≈ . σ j = ˜ σ ∗ + 4 π ≈ . b = 3 BC the FBMS obtained by maximizing the j = 3 and j = 5 eigenvalues are displayedin Figure 18. If Figures 16 and 17, the first few eigenfunctions are plotted in the optimal domains,Ω c,r . The eigenvalues obtained are ˜ σ = 23 . σ = 34 . b = 3 BC, to obtain eigenvalues ˜ σ =12 . · · π ≈ . σ = 12 . · · π ≈ . Discussion
In this paper, we developed computational methods to maximize the length-normalized j -thSteklov eigenvalue, ˜ σ j (Σ , g ) := σ j (Σ , g ) L ( ∂ Σ , g ) over the class of smooth Riemannian metrics, g ona compact surface, Σ, with genus γ and b boundary components. Our numerical method involves (i)using conformal uniformization of multiply connected domains to avoid explicit parameterizationfor the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem inmulti-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. Using the connection due to Fraser and Schoen [FS15], thesolutions to this extremal Steklov eigenvalue problem for various values of b boundary componentsare used to generate free boundary minimal surfaces.In hindsight, it may have been better to perform these computations on a punctured sphererather than a punctured disk, as a punctured disk distinguishes one boundary (the ‘outer’ one). Inparticular, by considering a punctured sphere, it may be that the holes appear more symmetricallythan for a punctured disk; see Figure 7.Beyond further exploring higher eigenvalues j and higher numbers of boundary components b , there are a number of interesting extensions of this work. In particular, we would be veryinterested to compute extremal Steklov eigenvalues on the M¨obius band, torus, and other higher enus surfaces and use the associated eigenfunctions to generate free boundary minimal surfaces.We’re also interested in related extremal eigenvalue problems, involving convex combinations ofSteklov eigenvalues or Steklov eigenvalues for the p -Laplacian. Acknowledgements.
The authors would like to thank the Mathematics Division, National Centerof Theoretical Sciences, Taipei, Taiwan for hosting a research pair program during June 15-June 30,2019 to support this project. Chiu-Yen Kao acknowledges partial support from NSF DMS 1818948.Braxton Osting acknowledges partial support from NSF DMS 17-52202. ´Edouard Oudet acknowl-edges partial support from CoMeDiC (ANR-15-CE40-0006) and ShapO (ANR-18-CE40-0013). Theauthors would also like to thank Bruno Colbois, Joel Dahne, Baptiste Devyver, Alexandre Girouard,Mikhail Karpukhin, Jean Lagace and Iosif Polterovich for useful conversations.
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E-mail address : [email protected] Department of Mathematics, University of Utah, Salt Lake City, UT
E-mail address : [email protected] LJK, Universit´e Grenoble Alpes, France
E-mail address : [email protected]@imag.fr