## Abstract

We consider a maximization problem for eigenvalues of the Laplace–Beltrami operator on surfaces of revolution in *j*, we show there is a surface *Σ*_{j} that maximizes the *j*th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.

## 1. Introduction

On a smoothly bounded planar domain *Ω*, the Dirichlet eigenvalues form a sequence
*Ω* is complicated. An interesting problem is to find domains that minimize eigenvalues subject to a geometric constraint. For example, Bucur *et al.* [1] showed that there is a domain that minimizes the second eigenvalue among bounded planar domains with the same perimeter. This was extended to higher eigenvalues by van den Berg & Iversen [2]. Bucur & Freitas [3] showed that these domains converge to a disc. These domains have been analysed numerically by Antunes & Freitas [4] and Bogosel & Oudet [5]. A minimization problem with more general constraints was considered by van den Berg [6].

In this article, we consider a related problem for surfaces in

Fix two distinct parallel circles *C*_{1} and *C*_{2} in *Γ*. Assume the circles are non-degenerate, i.e. have positive radius, and let *R*_{1} and *R*_{2} be the radii of *C*_{1} and *C*_{2}, respectively. There is an area-minimizing surface with boundary given by *C*_{1} and *C*_{2}. If the circles are coplanar, then a planar annulus is the unique area-minimizing surface. If the circles are not coplanar, then there are three cases, depending on the choice of the circles. In the first case, the union of two planar discs is the unique area minimizer. In the second case, there is a catenoid that is the unique area minimizer. In the third case, there are two area minimizers. One is the union of two planar discs and the other is a catenoid.

We pose an eigenvalue optimization problem in a similar spirit. Let *Σ*, satisfying the following three properties. First, *Σ* has two boundary components, given by the circles *C*_{1} and *C*_{2}. Second, *Σ* is disjoint from the axis *Γ*. Third, a meridian of *Σ* is an oriented rectifiable curve from *C*_{1} to *C*_{2}. Recall a meridian of *Σ* is the intersection of *Σ* and a half-plane with boundary given by the axis *Γ*. A rectifiable curve is a curve that admits a Lipschitz continuous parametrization. For a smooth surface of revolution *Σ* in *Δ*_{Σ} be the Laplace–Beltrami operator on *Σ*. The Dirichlet eigenvalues of −*Δ*_{Σ} form an increasing sequence,
*j*, define

If *C*_{1} and *C*_{2} are coplanar, then the planar annulus *A* in _{j}(*A*)=*Λ*_{j} for all *j*. This follows from [7], theorem 1.1 and an approximation argument, see lemma 3.1. If the circles *C*_{1} and *C*_{2} are not coplanar, then theorem 1.1 establishes the existence of eigenvalue-maximizing surfaces. Let *D*_{1} and *D*_{2} be the planar discs embedded in *C*_{1} and *C*_{2}, respectively. For each positive integer *j*, let λ_{j}(*D*_{1}∪*D*_{2}) be the *j*th Dirichlet eigenvalue of *D*_{1}∪*D*_{2}.

### Theorem 1.1

*Fix a positive integer j, and assume that Λ*_{j}*>λ*_{j}*(D*_{1}*∪D*_{2}*). Then, there is a surface Σ*_{j} *in* *such that λ*_{j}*(Σ*_{j}*)=Λ*_{j}.

The case *j*=1 of theorem 1.1 was established in [8]. Moreover, in this case, the maximizing surface was shown to be smooth. We prove theorem 1.1 in § 3, using an argument based on the Arzela–Ascoli theorem. In order to show that a maximizing sequence is contained in a compact set, the key step is to show that a maximizing sequence of surfaces is uniformly bounded between two cylinders about the axis *Γ*. This follows from an annulus comparison inequality, see lemma 3.2. This inequality was established in [7] for piecewise smooth surfaces of revolution, and we extend it to surfaces in

If there is a catenoid which is the unique area minimizing surface with boundary given by *C*_{1} and *C*_{2}, then theorem 1.2 shows that the eigenvalue-maximizing surfaces given by theorem 1.1 converge to this catenoid.

### Theorem 1.2

*Assume there is a catenoid Σ*_{cat} *which is the unique area-minimizing surface with boundary given by C*_{1} *and C*_{2}*. Then, Λ*_{j}*>λ*_{j}*(D*_{1}*∪D*_{2}*) for large j. For such j, let Σ*_{j} *be a surface in* *such that λ*_{j}*(Σ*_{j}*)=Λ*_{j}*. Then,
**Moreover, Σ*_{j} *converges to Σ*_{cat} *in the Hausdorff metric.*

We prove theorem 1.2 in §4, using an argument based on Weyl’s law. If *Σ* is a surface in *Λ*_{j}>λ_{j}(*D*_{1}∪*D*_{2}) for large *j* in theorem 1.2. The argument we use to prove theorem 1.2 yields a stronger result, see lemma 4.5. Namely, we can weaken the hypothesis that λ_{j}(*Σ*_{j})=*Λ*_{j} for large *j* and instead only assume that
*Σ*_{cat} is area minimizing. The proof of (1.7) relies crucially on the assumption that *Σ*_{cat} is the unique area-minimizing surface. This assumption enables us to prove that there are two cylinders of positive radius centred about the axis *Γ* such that each surface *Σ*_{j} is in the region bounded between these cylinders. Then, we can prove that the surfaces *Σ*_{j} have meridians with uniformly bounded length. For any sequence of surfaces in *Σ*_{j} converge to *Σ*_{cat} in the Hausdorff metric. We are not able to draw any conclusions when *D*_{1}∪*D*_{2} is an area minimizer. Specifically, we are not able to prove that the surfaces *Σ*_{j} are uniformly bounded between cylinders about the axis *Γ* or that their meridians have uniformly bounded length. Therefore, our proof of (1.7) does not apply, and we are not able to establish a statement analogous to (1.4) or any convergence of the eigenvalue-maximizing surfaces.

As described earlier, these results are closely related to the problem of minimizing Dirichlet eigenvalues among planar domains with fixed perimeter. Another related problem is to minimize Dirichlet eigenvalues among Euclidean domains of fixed volume. The Faber–Krahn inequality states that a ball minimizes the first Dirichlet eigenvalue among such sets. By the Krahn–Szegö inequality, the union of two balls of equal radii minimizes the second Dirichlet eigenvalue. Bucur & Henrot [9] showed that there is a quasi-open set that minimizes the third eigenvalue. This was extended to higher eigenvalues by Bucur [10]. A similar result for any increasing functional of finitely many eigenvalues was established by Mazzoleni & Pratelli [11].

Although eigenvalue-optimizing domains are often not known explicitly, there are other situations where the limit of eigenvalue optimizing domains is an explicit shape. Antunes & Freitas [12] showed that the rectangle of given area that minimizes the *j*th Dirichlet eigenvalue converges to a square. A similar result for the rectangle which maximizes the *j*th Neumann eigenvalue was established by van den Berg *et al.* [13]. Antunes & Freitas [4] showed that the rectangular parallelepiped that minimizes the *j*th Dirichlet eigenvalue subject to a surface area constraint on the boundary converges to a cube. The asymptotic behaviour of eigenvalue-minimizing domains is related to the Pólya conjecture. In two dimensions, Pólya’s conjecture states that if *Ω* is a planar domain with area one, then
*j*th Dirichlet eigenvalue among planar domains with volume one, then Colbois & El Soufi [14] showed that (1.8) is equivalent to

This article is particularly inspired by the work of Abreu & Freitas [15], who studied the optimization of eigenvalues of *et al.* [16] extended this to higher dimensions. These results were motivated by Hersch [17], who showed that on a sphere, the round metric maximizes the first non-zero eigenvalue among metrics of given area.

In §2, we define the eigenvalues λ_{j}(*Σ*) for any surface *Σ* in

## 2. Eigenvalues on rectifiable surfaces of revolution

In this section, we define the eigenvalues λ_{j}(*Σ*) for any surface *Σ* in *Σ* in _{j}(*Σ*) is well known. The main purpose of this section is to extend this definition to surfaces in

Fix an open half-plane in *Γ*. Identify this half-plane with *p* and *q* be the points, where *C*_{1} and *C*_{2}, respectively. Let *α*(0)=*p* and *α*(1)=*q*. Second, there is a constant *L* such that |*α*′(*t*)|=*L* for almost every *t*. Note that there is a bijective correspondence between

Before defining the eigenvalues for surfaces in *Σ* is a smooth surface in *α* be the corresponding smooth curve in _{j}(*α*)=λ_{j}(*Σ*) for each *j*. Let *Lip*_{0}(*Σ*) be the space of functions *Σ*. Then for each *j*,
*j*-dimensional subspaces *V* of Lip_{0}(*Σ*). In addition, ∇ is the Riemannian gradient and d*S* is the Riemannian measure on *Σ*. Separation of variables shows that every eigenvalue can be realized by an eigenfunction that is the product of a radially symmetric function and a sinusoidal function. Therefore, we may express the eigenvalues in an alternative way. Write *α*=(*F*,*G*), i.e. let *F* and *G* be the component functions of *α*. Let *Lip*_{0}(0,1) be the space of functions *k* and a positive integer *n*, define
*n*-dimensional subspaces *W* of *Lip*_{0}(0,1). Then,
_{k,n}(*α*) twice for *k*≠0, the multiplicities agree.

Note that the right-hand side of (2.3) makes sense if the smooth curve *α* is replaced with a Lipschitz continuous curve in *α* in _{k,n}(*α*) for non-negative *k* and positive *n* by (2.3). It is well known that if *α* is smooth, then there are only finitely many eigenvalues λ_{k,n}(*α*) in any bounded subset of *α* is a Lipschitz continuous curve in *α* with the eigenvalues of a cylinder. Hence, if *α* is a Lipschitz continuous curve in _{j}(*α*) for positive integers *j* as follows. Counting the eigenvalue λ_{k,n}(*α*) twice for *k*≠0, we define λ_{j}(*α*) for positive integers *j*, so that (1.2) and (2.4) hold, with multiplicities agreeing in (2.4). Let *Σ* be the low regularity surface in *α*, and define λ_{k,n}(*Σ*)=λ_{k,n}(*α*) for all *k* and *n*. Likewise, for all *j*, define
*Σ* by

These definitions of eigenvalues also make sense for a slightly more general class of curves. Let [*c*,*d*] be an interval and let *δ*>0 such that |*α*′(*t*)|≥*δ* for almost every *t* in [*c*,*d*], then we may define the eigenvalues λ_{k,n}(*α*) and λ_{j}(*α*) similarly. For *k* fixed, the eigenvalues λ_{k,n}(*α*) are the eigenvalues of the regular Sturm–Liouville problem
*w*(*c*)=*w*(*d*)=0. This observation yields a continuity result for the functionals λ_{k,n}. To state this, assume there is a constant *L*>0 such that |*α*′(*t*)|=*L* for almost every *t* in [*c*,*d*]. For positive integers *m*, let *α*_{m}=(*F*_{m},*G*_{m}). Assume *F*_{m} converges to *F* uniformly over [*c*,*d*] and assume |*α*_{m}′| converges in *L*. For every *k* and *n*

## 3. Existence of maximizers

In this section, we prove theorem 1.1. In lemma 3.1, we prove that if a surface in *Γ*. In lemma 3.3, we bound the length of a surface in

### Lemma 3.1

*Let* *be Lipschitz. Assume there is a constant* *L* *such that* |*α*′(*t*)|=*L* *for almost every* *t* *in* [*c*,*d*]. *Write* *α*=(*F*_{α},*G*_{α}). *Let* [*ρ*_{1},*ρ*_{2}] *be the image of* *F*_{α}, *and assume* *ρ*_{1}<*ρ*_{2}. *Let* *A* *be an annulus in* *of radii* *ρ*_{1} *and* *ρ*_{2}. *Then*, λ_{k,n}(*α*)≤λ_{k,n}(*A*) *for every* *k* *and* *n*.

### Proof.

There is a subinterval of [*c*,*d*] such that *α* attains its minimum and maximum values at the endpoints of the subinterval. By the domain monotonicity of Dirichlet eigenvalues, the eigenvalues of the restriction of *α* to this subinterval are greater than or equal to the eigenvalues of *α*. Passing to the restriction, it suffices to consider the case where *F*_{α} attains its minimum and maximum values *ρ*_{1} and *ρ*_{2} at the endpoints *c* and *d*. Furthermore, by a change of variables, we may assume that the domain [*c*,*d*] is equal to [0,1]. Note that if *α* is smooth and regular, then λ_{k,n}(*α*)≤λ_{k,n}(*A*) for every *k* and *n* by [7], lemma 2.1. First, we use an approximation argument to extend these inequalities to continuously differentiable curves. Then, we use another approximation argument to extend these inequalities to Lipschitz curves.

For now assume that the curve *L* such that |*α*′(*t*)|=*L* for every *t* in [*c*,*d*]. Define *γ* is continuously differentiable over [−1,2], with |*γ*′(*t*)|=*L* for every *t* in [−1,2]. Write *γ*=(*F*_{γ},*G*_{γ}). For *ε*>0, let *F*_{ε} and *G*_{ε} be the standard mollifications of *F*_{γ} and *G*_{γ}, respectively, see e.g. [19], pp. 122–123. Restrict the domains of *F*_{ε} and *G*_{ε} to [0,1]. Then, *F*_{ε} and *G*_{ε} are smooth. Moreover, *F*_{ε} and *G*_{ε} converge uniformly in *C*^{1} to *F*_{α} and *G*_{α}, respectively. Define *γ*_{ε}=(*F*_{ε},*G*_{ε}). Then by (2.8),
*ρ*_{1}≤*F*_{ε}(*t*)≤*ρ*_{2} for all *ε*>0 and all *t* in [0,1]. Furthermore, *F*_{ε}(0)=*ρ*_{1} and *F*_{ε}(1)=*ρ*_{2}. Note that λ_{k,n}(*γ*_{ε})≤λ_{k,n}(*A*) for every *ε*, *k* and *n* by the result mentioned above, because *γ*_{ε} is a smooth regular curve and the image of *F*_{ε} is [*ρ*_{1},*ρ*_{2}]. Therefore, λ_{k,n}(*α*)≤λ_{k,n}(*A*) for every *k* and *n*. This completes the proof for the case where *α* is continuously differentiable.

We complete the proof, using another approximation argument. Now assume that the curve *L* such that |*α*′(*t*)|=*L* for almost every *t* in [0,1]. The argument differs from the previous one in the definition of *G*_{ε}. Define *γ*=(*F*_{γ},*G*_{γ}). For *ε*>0, let *F*_{ε} be the standard mollification of *F*_{γ}. Restrict the domain of *F*_{ε} to [0,1]. Then, *F*_{ε} is smooth and |*F*_{ε}′(*t*)|≤*L* for every *t* in [0,1]. Also the image of *F*_{ε} is [*ρ*_{1},*ρ*_{2}]. Define *G*_{ε}(0)=*G*_{α}(0) and for all *t* in [0,1],
*γ*_{ε}=(*F*_{ε},*G*_{ε}). Then, *γ*_{ε} is continuously differentiable and |*γ*_{ε}′(*t*)|=*L* for all *t* in [0,1]. Moreover, *F*_{ε} converges to *F*_{α} uniformly over [0,1] as *ε*→0. By (2.8),
_{k,n}(*γ*_{ε})≤λ_{k,n}(*A*) for every *ε*, *k* and *n*, by the argument above, because *γ*_{ε} is a continuously differentiable curve such that |*γ*_{ε}′(*t*)|=*L* for every *t* in [0,1], and the image of *F*_{ε} is [*ρ*_{1},*ρ*_{2}]. Therefore, λ_{k,n}(*α*)≤λ_{k,n}(*A*) for every *k* and *n*. This completes the proof. ▪

We use lemma 3.1 to show that if a surface in *Γ*. Recall that *R*_{1} and *R*_{2} are the radii of *C*_{1} and *C*_{2}, respectively.

### Lemma 3.2

*Fix constants* *b*>*a*>0. *Assume* *a* *is less than* *R*_{1} *and* *R*_{2}. *In addition, assume* *b* *is greater than* *R*_{1} *and* *R*_{2}. *Let* *A*_{1} *and* *A*_{2} *be disjoint annuli in* *with inner radii* *a* *and outer radii* *R*_{1} *and* *R*_{2}, *respectively. Let* *B*_{1} *and* *B*_{2} *be disjoint annuli in* *with outer radii* *b* *and inner radii* *R*_{1} *and* *R*_{2}, *respectively. Let* *α* *be a curve in* *and write* *α*=(*F*,*G*). *If* λ_{j}(*α*)>λ_{j}(*A*_{1}∪*A*_{2}) *for some* *j*, *then* *F*(*t*)≥*a* *for all* *t* *in* [0,1]. *Likewise, if* λ_{j}(*α*)>λ_{j}(*B*_{1}∪*B*_{2}) *for some* *j*, *then* *F*(*t*)≤*b* *for all* *t* *in* [0,1].

### Proof.

Assume that λ_{j}(*α*)>λ_{j}(*A*_{1}∪*A*_{2}) for some *j*. Fix *t*_{0} in [0,1] and suppose that *F*(*t*_{0})<*a*. Note that *t*_{0}≠0 and *t*_{0}≠1, because *α* is in *β* and *γ* be the restrictions of *α* to [0,*t*_{0}] and [*t*_{0},1], respectively. Then, λ_{k,n}(*β*)≤λ_{k,n}(*A*_{1}) for all *k* and *n*, by lemma 3.1 and by the domain monotonicity of Dirichlet eigenvalues. Hence, λ_{j}(*β*)≤λ_{j}(*A*_{1}) for all *j*. Likewise, λ_{j}(*γ*)≤λ_{j}(*A*_{2}) for all *j*. Therefore, λ_{j}(*α*)≤λ_{j}(*A*_{1}∪*A*_{2}) for all *j*. This contradiction shows that *F*(*t*)≥*a* for all *t* in [0,1]. A similar argument can be used to prove that if λ_{j}(*α*)>λ_{j}(*B*_{1}∪*B*_{2}) for some *j*, then *F*(*t*)≤*b* for all *t* in [0,1]. ▪

Lemma 3.3 provides an upper bound for eigenvalues of a curve in terms of the length.

### Lemma 3.3

*Let* *α* *be a curve in* *Write* *α*=(*F*,*G*), *and let* *L* *be the length of* *α*. *Assume that there are positive constants* *a* *and* *b* *such that* *a*≤*F*(*t*)≤*b* *for every* *t* *in* [0,1]. *Fix* *j*. *Then*,

### Proof.

For each *i*=1,2,…,*j*, define a function *w*_{i} in *Lip*_{0}(0,1) by
*i*,
*W* be the *j*-dimensional subspace of *Lip*_{0}(0,1) generated by the functions *w*_{i}. Because the supports of these functions are disjoint, it follows from (3.7) that
_{j}(*α*)≤λ_{0,j}(*α*), this completes the proof. ▪

Now, we can prove theorem 1.1.

### Proof of theorem 1.1.

Fix *j*. Fix constants *b*>*a*>0. Assume *a* is less than *R*_{1} and *R*_{2}. Also assume *b* is greater than *R*_{1} and *R*_{2}. Let *A*_{1} and *A*_{2} be disjoint annuli in *a* and outer radii *R*_{1} and *R*_{2}, respectively. Let *B*_{1} and *B*_{2} be disjoint annuli in *b* and inner radii *R*_{1} and *R*_{2}, respectively. Recall that *Λ*_{j}>λ_{j}(*D*_{1}∪*D*_{2}), by assumption. Therefore, if *a* is small, then *Λ*_{j}>λ_{j}(*A*_{1}∪*A*_{2}). For a proof of this, we refer to Rauch & Taylor [20], who considered a much more general problem. If *b* is large, then *Λ*_{j}>λ_{j}(*B*_{1}∪*B*_{2}). Let *Σ*_{i} be a sequence in _{j}(*Σ*_{i})>λ_{j}(*A*_{1}∪*A*_{2}) and λ_{j}(*Σ*_{i})>λ_{j}(*B*_{1}∪*B*_{2}) for every *i*. Let *α*_{i} be the curve in *Σ*_{i}. Write *α*_{i}=(*F*_{i},*G*_{i}). Then, *a*≤*F*_{i}(*t*)≤*b* for every *t* in [0,1] and every *i*, by lemma 3.2. Let *L*_{i} be the length of *α*_{i}. By lemma 3.3, the lengths *L*_{i} are uniformly bounded. By passing to a subsequence, we may assume that there is a *L*>0 such that the lengths *L*_{i} converge to *L*. By applying the Arzela–Ascoli theorem and passing to a subsequence, we may assume that the curves *α*_{i} converge uniformly to a curve *α*′(*t*)|≤*L* for almost every *t* in [0,1]. Write *α*=(*F*,*G*). Then *F*:[0,1]→[*a*,*b*] satisfies |*F*′(*t*)|≤*L* for almost every *t* in [0,1]. There is a Lipschitz function *t* in [0,1],
*H*′(*t*)|≥|*G*′(*t*)| for almost every *t* in [0,1]. Moreover, we may choose the function *H*, so that *H*(0)=*G*(0) and *H*(1)=*G*(1). Define *β*=(*F*,*H*). Then, *β* is in *k* and *n*,
*Σ*_{j} be the surface in *β*. Then
*Σ*_{j} is in _{j}(*Σ*_{j})=*Λ*_{j}. ▪

## 4. Convergence to the catenoid

In this section, we prove theorem 1.2. In fact, we prove a slightly stronger statement, see lemma 4.5. In lemma 4.1, we show that a sequence of surfaces with large eigenvalues is uniformly bounded between two cylinders about the axis *Γ*. In lemma 4.2, we show that such a sequence of surfaces have uniformly bounded length. Lemma 4.3 provides bounds for eigenvalues on rectangles, which we later use to establish (1.7). In lemma 4.4, we show that surfaces with small area must approximate the minimizing catenoid. Then, we prove lemma 4.5, which establishes theorem 1.2.

### Lemma 4.1

*Assume there is a catenoid* *Σ*_{cat} *which is the unique area-minimizing surface with boundary given by* *C*_{1} *and* *C*_{2}. *Let* *α*_{j} *be curves in* *such that*
*Write* *α*_{j}=(*F*_{j},*G*_{j}). *Then, there are positive constants* *a* *and* *b* *such that* *a*≤*F*_{j}(*t*)≤*b* *for large* *j* *and every* *t* *in* [0,1].

### Proof.

Fix constants *b*>*a*>0. Assume *a* is less than *R*_{1} and *R*_{2}. Also assume *b* is greater than *R*_{1} and *R*_{2}. Let *A*_{1} and *A*_{2} be disjoint annuli in *a* and outer radii *R*_{1} and *R*_{2}, respectively. Let *B*_{1} and *B*_{2} be disjoint annuli in *b* and inner radii *R*_{1} and *R*_{2}, respectively. If *a* is small and *b* is large, then we have *Area*(*Σ*_{cat})<*Area*(*A*_{1}∪*A*_{2}) and *Area*(*Σ*_{cat})<*Area*(*B*_{1}∪*B*_{2}). Then by Weyl’s law,
_{j}(*α*_{j})>λ_{j}(*A*_{1}∪*A*_{2}) and λ_{j}(*α*_{j})>λ_{j}(*B*_{1}∪*B*_{2}) for large *j*. Therefore, *a*≤*F*_{j}(*t*)≤*b* for large *j* and for every *t* in [0,1], by lemma 3.2. ▪

Next, we show that the lengths of the eigenvalue maximizing curves are uniformly bounded. We prove this by comparing the eigenvalues to those of a cylinder.

### Lemma 4.2

*Assume there is a catenoid* *Σ*_{cat} *which is the unique area-minimizing surface with boundary given by* *C*_{1} *and* *C*_{2}. *Let* *α*_{j} *be curves in* *such that*
*Then, the lengths* *L*_{j} *of the curves* *α*_{j} *are uniformly bounded*.

### Proof.

Write *α*_{j}=(*F*_{j},*G*_{j}). There are positive constants *a* and *b* such that *a*≤*F*_{j}(*t*)≤*b* for large *j* and every *t* in [0,1], by lemma 4.1. Therefore, if *j* is large, then for any function *w* in *Lip*_{0}(0,1) and any *k*,
*C*_{j} be a cylinder of radius *a* and height *L*_{j}. By (4.5), if *j* is large, then for any *k* and *n*,
*j* is large,
*L*_{j} are unbounded. Let *M*>0. Then, there are infinitely many *j* such that *L*_{j}≥*M*. Let *C*_{M} be a cylinder of radius *a* and height *M*. For all *j* such that *L*_{j}≥*M*, the domain monotonicity of Dirichlet eigenvalues implies that
*M* may be arbitrarily large, this yields
*L*_{j} must be uniformly bounded. ▪

Lemma 4.3 provides bounds for Dirichlet eigenvalues on a union of rectangles. These estimates are useful because they only depend on the area and perimeter of the rectangles and are otherwise independent of the choice of rectangles. These estimates can also be used to bound Dirichlet eigenvalues on cylinders. In the proof of theorem 1.2, we apply these estimates to obtain bounds for the eigenvalues λ_{j}(*Σ*_{j}) in terms of the area of *Σ*_{j}, in order to establish (1.7).

### Lemma 4.3

*Let* *Q*_{1},…*Q*_{N} *be disjoint compact rectangles in* *Define* *Q*=∪*Q*_{i}. *For every* *j*, *such that* λ_{j}(*Q*)>1,

### Proof.

For each *m*=1,2,…,*N*, let *D*(λ,*Q*_{m}) be the number of Dirichlet eigenvalues on *Q*_{m} which are less than or equal to λ. For every λ>0 and every *m*, it is well known that
*D*(λ,*Q*) similarly, and note that
*m* in (4.12) yields
*j* such that λ_{j}(*Q*)>1 and set λ=λ_{j}(*Q*)−1. Then, *j*≥*D*(λ_{j}(*Q*)−1,*Q*), so (4.11) follows. ▪

If there is a catenoid which is the unique area minimizer, then lemma 4.4 shows that surfaces in

### Lemma 4.4

*Assume there is a catenoid* *Σ*_{cat} *which is the unique area minimizing surface with boundary given by* *C*_{1} *and* *C*_{2}. *Let* *Σ*_{j} *be a sequence of surfaces in* *such that*
*Then, the surfaces* *Σ*_{j} *converge to* *Σ*_{cat} *in the Hausdorff metric*.

### Proof.

Fix constants *b*>*a*>0. Assume *a* is less than *R*_{1} and *R*_{2}. In addition, assume *b* is greater than *R*_{1} and *R*_{2}. Let *A*_{1} and *A*_{2} be disjoint annuli in *a* and outer radii *R*_{1} and *R*_{2}, respectively. Let *B*_{1} and *B*_{2} be disjoint annuli in *b* and inner radii *R*_{1} and *R*_{2}, respectively. If *a* is small and *b* is large, then we have *Area*(*Σ*_{cat})<*Area*(*A*_{1}∪*A*_{2}) and *Area*(*Σ*_{cat})<*Area*(*B*_{1}∪*B*_{2}). Then, we may assume that for every *j*,

To prove convergence of the surfaces *Σ*_{j}, we show that any subsequence admits a subsequence which converges to *Σ*_{cat} in the Hausdorff metric. Let *Σ*_{jk} be an arbitrary subsequence. For each *k*, let *α*_{k} be the curve in *Σ*_{jk}. Write *α*_{k}=(*F*_{k},*G*_{k}), and let *L*_{k} be the length of *α*_{k}. Note that (4.16) and (4.17) imply that *a*≤*F*_{k}(*t*)≤*b* for every *k* and every *t* in [0,1]. It then follows that the lengths *L*_{k} are uniformly bounded. By passing to a subsequence, we may assume that the lengths *L*_{k} converge to some positive constant *L*. By the Arzela–Ascoli theorem, there is a subsequence *α*_{kn} which converges uniformly to some Lipschitz curve *β*′(*t*)|≤*L* for almost every *t* in [0,1]. Write *β*=(*F*_{β},*G*_{β}). Now,
*Σ*_{β} be the surface in *β* parametrizes a meridian of *Σ*_{β}. Then, by (4.15) and (4.18),
*Σ*_{cat} is the unique area minimizer, this implies that *Σ*_{β}=*Σ*_{cat}, i.e. *β* parametrizes a meridian of *Σ*_{cat}. Now the uniform convergence of the curves *α*_{kn} to *β* implies that the surfaces in *α*_{kn} converge to *Σ*_{cat} in the Hausdorff metric. That is, a subsequence of *Σ*_{jk} converges to *Σ*_{cat} in the Hausdorff metric. Therefore, the full sequence of surfaces *Σ*_{j} converge to *Σ*_{cat} in the Hausdorff metric. ▪

Now, we conclude the article by proving theorem 1.2. Because *Σ*_{cat} is in

### Lemma 4.5

*Assume there is a catenoid* *Σ*_{cat} *which is the unique area minimizing surface with boundary given by* *C*_{1} *and* *C*_{2}. *Let* *Σ*_{j} *be a sequence of surfaces in* *such that*
*Then*,
*Moreover*, *Σ*_{j} *converges to* *Σ*_{cat} *in the Hausdorff metric*.

### Proof.

For each *j*, let *α*_{j} be the curve in *Σ*_{j}, and let *L*_{j} be the length of *α*_{j}. Let 0<*ε*<1. Let *N*>0 be an integer, and partition [0,1] into *N* subintervals each of length 1/*N*. That is, for *m*=1,2,…,*N*, define
*α*_{j,m} be the restrictions of *α*_{j} to *I*_{m} for each *m*=1,2,…,*N*. Write *α*_{j,m}=(*F*_{j,m},*G*_{j,m}). Define *r*_{j,m} to be the minimum of *F*_{j,m}. Note that the maximum of *F*_{j,m} is at most *r*_{j,m}+*L*_{j}/*N*. Moreover, by lemmas 4.1 and 4.2, the quantities *r*_{j,m} and *L*_{j} are uniformly bounded above and below by positive constants, independent of *j* and *m*. These bounds are also independent of *N*. In particular, we may assume that *N* is large, so that for every *j* and *m*,

Therefore, for every *j*, *k* and *m*, and for every *w* in *Lip*_{0}(*I*_{m}),
*C*_{j,m} be a cylinder of radius *r*_{j,m} and height *L*_{j}/*N*. Then, by (4.25), for all *j*, *k*, *m* and *n*,
*j* and *m*, let *Q*_{j,m} be a rectangle of width 2*πr*_{j,m} and height *L*_{j}/*N*. Assume the rectangles *Q*_{j,m} are disjoint, and define
_{j}(*C*_{j,m})≤λ_{j}(*Q*_{j,m}) for every *j* and *m*, so (4.26) yields
*j*,

Because of the uniform bounds on *r*_{j,m} and *L*_{j}, there is a constant *C*_{N} such that Perimeter(*Q*_{j})≤*C*_{N} for every *j*. Therefore, if *j* is large, then by lemma 4.3,
*ε*<1 is arbitrary and *Σ*_{cat} is area minimizing, this yields
*Σ*_{j} converge to *Σ*_{cat} in the Hausdorff metric. ▪

## Competing interests

The author has no competing interests.

## Funding

The author was supported by CAPES and IMPA of Brazil through the programme Pós-Doutorado de Excelência.

- Received April 1, 2016.
- Accepted September 6, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.