A necessary condition for the existence of a doubly connected minimal surface

If we dip a wire contour into a solution of a soapy water and then pull it out, the soap film formed by the frame is a surface with the least possible area among all surfaces having the same contour. The energy of the soap film is due to the forces of attraction between the molecules (surface tension) and where the gravity is neglected (small wire contours). Thus this energy is proportional to the area of the soap film. Mathematically, a surface that locally minimizes the area has the property that its mean curvature is zero everywhere and it is called a minimal surface.

In this paper, we are interested in the case that the wire contour is formed by two circles \(C_1\) and \(C_2\) and it is motivated by the experiments that one can easily do with the soap films framed by two circles. Firstly, we dip two coaxial circles \(C_1\cup C_2\) in parallel planes \(\varPi _1\cup \varPi _2, C_i\subset \varPi _i, i=1,2\). If \(C_1\) is sufficiently close to \(C_2\), the soap film obtained is a catenoid, which is, besides the plane, the only rotational minimal surface. From early works of Plateau, Goldschmidt and Lindelöf, it is known that after a critical value of the vertical distance between the circles, the catenoid breaks into two disks [2, 4, 14]. In fact, and depending on the distance h between the two circles, there are zero, one or two catenoids, one of which is unstable in the latter case. For example, if \(C_1\) and \(C_2\) have the same radius r, there exists \(h_0 \simeq 1.325 r\) such that if \(h<h_0\), there are two catenoids spanning \(C_1\) and \(C_2\) and only one is physically realizable; if \(h=h_0\) there is exactly one catenoid bounded by \(C_1\cup C_2\); and if \(h>h_0\), there is not a catenoid joining \(C_1\) with \(C_2\).

left: A piece of a Riemann minimal example. right: the surface is obtained by the reflection across the plane of equation \(z=0\) of the piece on the left side. Here \(a=4\) and \(b=2\)

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Once that we have formed a catenoid we now displace slightly the circles \(C_1\) and \(C_2\) sideways in a direction parallel to \(\varPi _i\) without breaking the soap film. In such a case, we go obtaining a family of minimal surfaces with the topology of an annulus, also called in topology, a doubly connected surface. Shiffman proved in [17] that a minimal annulus spanning two circles in parallel planes is, indeed, foliated by circles in parallel planes, that is, the intersection of the surface with a parallel plane to \(\varPi _i\) is a circle. This surface belongs to a family of minimal surfaces discovered by Riemann in 1860s and called in the literature a Riemann minimal example [15]. A Riemann minimal example is a surface with zero mean curvature constructed by a uniparametric family of circles in parallel planes. See Fig. 1. Recall here that Enneper proved that if a minimal surface is foliated by a uniparametric family of circles, then the foliating planes must be parallel [1] and thus, the surface is rotational or it is one of the Riemann minimal examples. More properties of these surfaces will appear in Sect. 2.

After constructing a catenoid, and then a piece of a Riemann minimal example, we follow by displacing \(C_1\) and \(C_2\) sideways by keeping the vertical distance h. If \(C_1\) and \(C_2\) are sufficiently far, the surface breaks into two disks, namely the two disks bounded by each circle \(C_i\) in \(\varPi _i\). If d denotes the lateral distance between the centers of \(C_1\) and \(C_2\), the above experiments can summarize as follows. If \(d=0\), the surface is the catenoid. For \(d>0\) close to 0, we obtain a Riemann minimal example and if d is sufficiently big, the surface leaves to be connected. It is a problem in classical theory of minimal surfaces to estimate the value \(d_1=d_1(r_1,r_2,h)\) such that no doubly connected minimal surface bounded by \(C_1\) and \(C_2\) exists for \(d>d_1\). The control of the number \(d_1(h)\) appears as ‘Problem 33’ in [8]. A similar question can be considered for two Jordan curves \(\varGamma _1\) and \(\varGamma _2\) contained in \(\varPi _1\) and \(\varPi _2\), respectively. We point out the main results related with this paper.

1. 1.

   If there exists a connected minimal surface bounded by \(C_1\cup C_2\), then the orthogonal projection on a plane parallel to \(\varPi _i\) of \(C_1\) and \(C_2\) must overlap [10]. This result is more general, and it is known that if there exists a plane orthogonal to \(\varPi _1\cup \varPi _2\) separating \(C_1\) and \(C_2\), then \(C_1\cup C_2\) cannot bound a minimal surface of annulus type (see also [16] when \(C_1\) and \(C_2\) are convex curves by using an argument with the touching principle of minimal surfaces).

2. 2.

   Denote by \(\delta _i\) the diameter of \(\varGamma _i\). If there exists a (connected) minimal surface, then \(h\le \max \{\delta _1,\delta _2\}\) ([16]). In particular, the surface is contained in a vertical slab of width \(\max \{\delta _1,\delta _2\}\). This generalizes previous works of Nitsche for doubly connected surfaces [6].

3. 3.

   For a doubly connected minimal surface we have:

   $$\begin{aligned} h\le \sqrt{\frac{\left( \delta _1+\delta _2\right) ^2}{4}-\frac{d^2}{2}}. \end{aligned}$$

   (1)

   See [7]. Inequality (1) is generalized in [13] for surfaces in \(\mathbb R^n\) without requiring that \(\varGamma _1\) and \(\varGamma _2\) lie in parallel hyperplanes.

4. 4.

   There does not exist a doubly connected minimal surface if \(\varGamma _1\) and \(\varGamma _2\) lie in different components of the cone \(x^2+y^2<z^2\sinh ^2\tau \), where \(\tau \) is the unique positive solution of \(\cosh \tau -\tau \sinh \tau =0\): see [13], extending previous results of Hildebrandt [3].

Motivated by these results, we pose the next two problems:

1. 1.

   Problem 1. By moving \(C_1\) and \(C_2\) sideways along a fix direction, can we displace \(C_1\) exactly until just before the orthogonal projection of \(C_1\) onto \(\varPi _2\) is tangent to \(C_2\) obtaining during this displacement a Riemann minimal example?

2. 2.

   Problem 2. Determine a function \(M(r_1,r_2,d)\) such that the curves \(\varGamma _1\) and \(\varGamma _2\) cannot bound a doubly connected minimal surface whenever \(h>M(r_1,r_2,d)\). This appears as ‘Theorem’ in [12].

In Sect. 2 we will give a universal bound for the overlapping distance of the circles \(C_1\) and \(C_2\) in such a way that if this distance is bigger than this bound, then there does not exist a doubly connected minimal surface spanning \(C_1\cup C_2\) (Theorem 1). This result generalizes for two Jordan curves in parallel planes. In Sect. 3 we compare the height of a piece of a Riemann minimal example bounded by two circles with the one of a catenoid with the same boundary radii and waist. Finally in Sect. 4, numerical results are presented that will indicate the approximation of the proposed estimate. Motivated by the experiments, the calculations fix the radii of the circles as well as the vertical distance.

We review the description of the Riemann minimal examples. Here, we follow [11]. Let (x, y, z) be the standard coordinates in Euclidean space \(\mathbb R^3\), where (x, y) indicate the horizontal coordinates and z is the vertical direction. Let \(\varPi \) be the plane of equation \(z=0\). A Riemann minimal example S foliated by circles in parallel horizontal planes is parametrized as \(X(u,\theta )=(c(u),0,u)+r(u)(\cos \theta ,\sin \theta ,0)\), where (c(u), 0, u)), \(u\in I\subset \mathbb R\), is the planar curve of centers of the circles. The minimality of S writes as

for all \(u\in I\) and \(\theta \in \mathbb R\). Hence, we have two differential equations, namely

We point out that c is not a constant function because on the contrary S would be a catenoid (a solution of \(rr''=1+r'^2\)). Since \(c'\not =0\), the first equation gives \(c'=\lambda r^2\) for some positive constant \(\lambda \) and it follows that the second equation is \(-rr''+1+r'^2+\lambda ^2 r^4=0\). After some manipulations, there exists \(\mu \in \mathbb R\) such that

A Riemann minimal surface bounded by the circles \(C_1\cup C_2\). We indicate by h and d the vertical and the lateral distance between \(C_1\) and \(C_2\). The circle of the foliation of minimum radius is the waist

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By changing the variable u by the parameter radius r, the parametrization of S is

Let

Up to reversing the values of \(a^2\) and \(b^2\) if necessary, each circle of the foliation of radius u (\(u\ge b\)) at height z(u) is characterized in terms of elliptic integrals by

where

The parametrization of S is

where \(u\ge b\) and \(\theta \in \mathbb R\). The parametrization (4) is obtained when we choose the positive branch of the square root in (2) and corresponds with the part of the surface above the plane \(\varPi \). In the half-space \(z\le 0\), the choice in (2) is the negative branch, and the surface parametrizes as

Denote by \({\mathcal {R}}={\mathcal {R}}(a,b)\) the Riemann minimal example parametrized by (4) and (5). Some properties of \({\mathcal {R}}\) are:

1. 1.

   The intersection of \({\mathcal {R}}(a,b)\) with the plane \(\varPi \) is the circle of the foliation of minimum radius \(r=b\), called the waist of \({\mathcal {R}}(a,b)\). See Fig. 2.

2. 2.

   The surface is symmetric about the origin of coordinates and by a homothety, we have \(\lambda {\mathcal {R}} (a,b)={\mathcal {R}}_{\lambda a,\lambda b}\) for \(\lambda >0\).

3. 3.

   The surface lies in a horizontal slab \(-z_0<z<z_0\) such that \(\lim _{r\rightarrow \infty }z(r)=z_0\). Thus as r tends to \(\infty \), the limit of circles converge to a straight line orthogonal to the xz-plane and by the Schwarz reflection principle, the surface can extend by rotating \(180^{\circ }\) about this line. Repeating this process and by successive reflections, we obtain a periodic embedded surface foliated by circles and, at a discrete set of heights, the intersection of the surface with horizontal planes is a straight line.

We will only work with compact pieces of a Riemann minimal example. We precise this definition.

Definition 1

A Riemann minimal surface is a compact sub-surface of a Riemann minimal example bounded by two circles.

In particular, we exclude that the surface can contain a straight line. We denote \({\mathcal {R}}_{1,2}={\mathcal {R}}_{1,2}(a,b)\) to indicate that the boundary is formed by the circles \(C_1\) and \(C_2\) of radii \(r_1\) and \(r_2\), respectively, with \(z(r_1)<z(r_2)\). In the symmetric case, that is, \(r=r_1=r_2\) (and consequently, \(z(r_1)=-z(r_2)<0\) and \(c(r_1)=-c(r_2)<0\), we stand for \({\mathcal {R}}_r\).

We know that the orthogonal projection onto \(\varPi \) of \(C_1\) and \(C_2\) must overlap. This property can express by saying that the lateral distance of \(C_1\) and \(C_2\) is less than the sum of their radii [11, pp. 88–89]. In this section, we want to estimate the amount of overlapping. We define the overlapping distance between \(C_1\) and \(C_2\) as

See Fig. 2. In the symmetric case, \({\mathcal {O}}_{1,2}\) is simply \(2(r-c(r))\). Denote by \(h=z(r_2)-z(r_1)\) and by \(d=c(r_2)-c(r_1)\) the vertical distance and the lateral distance between \(C_1\) and \(C_2\), respectively. Under this notation, the Problem 1 expresses by saying: if \(r_1\), \(r_2\) and h are fix, is it possible to have \({\mathcal {O}}_{1,2}(a,b)\rightarrow 0\) for all a and b?

Theorem 1

There exists a universal constant \(M=\sqrt{2}-1\) such that it holds

for any Riemann minimal surface \({\mathcal {R}}_{1,2}(a,b)\).

Proof

Consider the part of \({\mathcal {R}}(a,b)\) bounded by the waist and a circle C of radius \(r>0\) that lies above the plane \(\varPi \). Then \(c(r)>0\) and we have

The function \(t/(t+\sqrt{t^2+a^2})\) is increasing on t and then its minimum value is attained at \(t=b\). Thus (3) gives

If \({\mathcal {R}}_{1,2}\) has points in both sides of \(\varPi \), then we apply the above estimate for each part of \({\mathcal {R}}_{1,2}\), and finally we sum both estimates, obtaining (6).

Finally suppose now that \({\mathcal {R}}_{1,2}\) lies in one side of \(\varPi \), for example, above \(\varPi \). A similar argument yields

\(\square \)

The estimate (6) can view as a restriction on the vertical separation between two circles to bound a Riemann minimal surface. As we pointed out in the Introduction, it is known experimentally that if we move vertically \(C_1\) and \(C_2\) in opposite vertical directions, there exists a critical separation \(h_0=h_0(r_1,r_2,d)\) where the surface breaks into the two disks bounded by \(C_1\) and \(C_2\). By Theorem 1, we now can estimate the value \(h_0\).

Corollary 1

Given \(r_1, r_2, d>0\), then

Proof

If \({\mathcal {R}}_{1,2}(a,b)\) is Riemann minimal surface bounded by \(C_1\) and \(C_2\), then \(h\le (r_1+r_2-d)/M\). Since this holds for any \(a\ge b>0\), it follows the result. \(\square \)

We can extend Theorem 1 for a doubly connected minimal surface spanning two Jordan curves. Let \(\varGamma _1\) and \(\varGamma _2\) be two Jordan curves and let \(C_i\) be the circumscribed circle of radius \(r_i\) for \(\varGamma _i\), \(i=1,2\). We define the lateral distance d and the overlapping distance \({\mathcal {O}}_{1,2}\) between \(\varGamma _1\) and \(\varGamma _2\) to be the ones between the circles \(C_1\) and \(C_2\). In this context, we formulate Theorem 1 in terms of a nonexistence result.

Corollary 2

If

then the curves \(\varGamma _1\) and \(\varGamma _2\) cannot bound a doubly connected minimal surface.

Proof

By contradiction, suppose that S is a doubly connected minimal surface bounded by \(\varGamma _1\cup \varGamma _2\). By the inclusion theorem that appears in [9], it is proved that the circumscribed circles \(C_1\) and \(C_2\) also bound a minimal surface \(\tilde{S}\) of the type of the annulus. By the Shiffman’s theorem, \(\tilde{S}\) must be a Riemann minimal surface. However the same estimate (8) holds for \(\tilde{S}\), which it is a contradiction with (6). \(\square \)

This result gives some information on the function \(M(r_1,r_2,d)\) proposed by Nitsche [8, Sec. IV.1] and by Nitsche and Leavitt in [12]. From (8), we have \( M(r_1,r_2,d)\le (\sqrt{2}+1)(r_1+r_2-d)\).

We now compare the Nitsche’s estimate (1) with the bound (6) in Theorem 1. The overlapping distance writes in terms of d and \(r_i\) as \({\mathcal {O}}_{1,2}=r_1+r_2-d\). Then (1) gives two types of inequalities for \({\mathcal {O}}_{1,2}\), namely,

and

In particular, both estimates depend on h and d or on h and \(r_1+r_2\). However, the inequality (6) gives a universal lower bound of \({\mathcal {O}}_{1,2}\) in terms of the vertical and lateral distances. Recall that the estimate (1) was proved with techniques of complex analysis and strongly using that the surface has the topology of an annulus.

Consider a catenoid whose rotational axis is the z-axis and suppose that the waist \(\omega \) of the catenoid is a circle in the plane \(\varPi \) centered at the origin. Then the catenoid parametrizes as

Denote by \(\mathcal {C}_{r_1,r_2,\omega }\) the part of the catenoid bounded by two circles of radii \(r_1\) and \(r_2\). The height \(h_\mathrm{cat}(r_1,r_2,\omega )\) of \(\mathcal {C}_{r_1,r_2,\omega }\) is the distance between the two boundaries circles and this value is

If \(\mathcal {C}_{r_1,r_2,\omega }\) is a symmetric catenoid, then \(r_1=r_2=r\) and the height is \(h_\mathrm{cat}(r,\omega )=2\omega \ \mathrm { arc}\cosh {(r/\omega )}\). We compare the height \(h_\mathrm{cat}(r_1,r_2,\omega )\) of the catenoid \(\mathcal {C}_{r_1,r_2,\omega }\) with the one of a Riemann minimal surface with the same radii and waist.

Theorem 2

The height \(h(r_1,r_2)=z(r_2)-z(r_1)\) of a Riemann minimal surface \({\mathcal {R}}_{1,2}(a,b)\) satisfies:

1. 1.

   If \(0\le z(r_1)<z(r_2)\), then

   $$\begin{aligned} \frac{a}{\sqrt{r_2^2+a^2}} h_\mathrm{cat}(r_1,r_2,b)\le h(r_1,r_2)\le \frac{a}{\sqrt{r_1^2+a^2}}h_\mathrm{cat}(r_1,r_2,b). \end{aligned}$$
2. 2.

   If \(z(r_1)<0<z(r_2)\), then for \(i=1,2\) we have

   $$\begin{aligned} \frac{a}{2\sqrt{r_i^2+a^2}} h_\mathrm{cat}(r_i,b) \le |z(r_i)|\le \frac{a}{2\sqrt{b^2+a^2}} h_\mathrm{cat}(r_i,b). \end{aligned}$$
3. 3.

   In the symmetric case, we have

   $$\begin{aligned} \frac{a}{\sqrt{r^2+a^2}}h_\mathrm{cat}(r,b)\le h(r)\le \frac{a}{\sqrt{b^2+a^2}}h_\mathrm{cat}(r,b). \end{aligned}$$

   In particular, \(h< h_\mathrm{cat}(r,b)\).

The inequality \(h< h_\mathrm{cat}(r,b)\) can formulate by saying that the height of a symmetric Riemann minimal surface is strictly less than the height of a catenoid of the same boundary radius and waist. Experimentally this inequality says that if we have formed a catenoid by a soap film framed by two given circles \(C_1\cup C_2\) of the same radius and we displace \(C_1\) and \(C_2\) sideways to produce Riemann minimal surfaces, if we want to keep the same waist of the catenoid, we have to reduce the vertical distance between \(C_1\) and \(C_2\) of the Riemann minimal surface.

Proof

In a first case, suppose \(0\le z(r_1)<z(r_2)\). For \(b \le r_1\le t\le r_2\) we have \(r_1^2+a^2\le t^2+a^2\le r_2^2+a^2\) and this yields

and similarly,

The first item in Theorem 2 follows from the value z(r) in (3), the height of \({\mathcal {R}}_{1,2}(a,b)\), namely, \(z(r_2)-z(r_1)\) and the expression of \(h_\mathrm{cat}(r_1,r_2,b)\) in (9).

Suppose now \(z(r_1)<0<z(r_2)\). For the part of \({\mathcal {R}}_{1,2}(a,b)\) that lies above the plane \(\varPi \), we have

Then (3) gives the estimate for \(z(r_2)\) in the second item. For \(z(r_1)<0\) the argument is similar.

The third item is proved by letting \(r_1=r_2=r\) with \(z(r_1)=-z(r_2)\), and using the previous item. \(\square \)

With the same ideas than in Theorem 2, we can estimate the center of each circle of the foliation.

Corollary 3

For a Riemann minimal surface \({\mathcal {R}}(a,b)\), consider the center (c(r), 0, z(r)) of the circle of \({\mathcal {R}}(a,b)\) at height \(z=z(r)\). Then

Proof

Without loss of generality, we suppose \(z(r)\ge 0\), so \(c(r)\ge 0\). For each \(b\le t\le r\), we have \(b^2+a^2\le t^2+a^2\). Then the expression of c(r) in (3) gives

The other inequality for c(r) uses \(t^2+a^2\le r^2+a^2\). The estimate for |c(r)| is a consequence that \(a\ge b\). \(\square \)

The estimate (6) is not sharp as we see when \(d\rightarrow 0\), where the limit surface is a catenoid. In such a case, and when the two circles have the same radius r, we have \({\mathcal {O}}_{1,2}=2r\), and thus \({\mathcal {O}}_{1,2}\ge 1.509 h\), in contrast to the estimate (6) that gives \({\mathcal {O}}_{1,2}\ge 0.4142 h\). However, the importance of our estimate is based that it holds for any radii \(r_1\) and \(r_2\) and any vertical distance h. The aim of this section is showing some computations of \({\mathcal {O}}_{1,2}\) in order to measure how amount of sharpness is the bound \(M=\sqrt{2}-1\). We also give some computations for the bound (7) of the vertical distance given in Corollary 1.

We begin with the estimate (6). Consider the symmetric case, that is, a Riemann minimal surface \({\mathcal {R}}_r\) bounded by two circles with the same radius \(r>0\). After an homothety of the space, we assume that the vertical distance is \(h=2\). Thus the value of z(r) is \(z(r)=1\). For this value of h, the estimate (6) becomes \({\mathcal {O}}_{1,2}\ge 0.82843\). We explain the steps to follow in the next computations.

1. 1.

   Fix the value of the radius r.

2. 2.

   For each value of the parameter a, compute the value of z(r) in (3) depending on the parameter b until that we get \(z(r)=1\).

3. 3.

   The equality \(z(r)=1\) provides the value of the waist b.

4. 4.

   Compute the x-coordinate c(r) of the center of the circle \(C_2\) by using (3).

5. 5.

   Compute the overlapping distance \({\mathcal {O}}_{1,2}=2(r-c(r))\).

Here we use the software Mathematica where the calculations have an approximation of 5 decimal digits.

In the above scheme, there appears the problem of whether any given circle \(C_2\) and the corresponding symmetric circle \(C_1\) at the height \(-z(r)\), bound a symmetric Riemann minimal surface \({\mathcal {R}}_r\). In a first step, we begin for small values for r, namely, \(r=3\) and \(r=5\) (Table 1). In each step of computation, and once fixed the value of a, we do not know a priori if there exists a value of b so the height of \({\mathcal {R}}_r\) is 2.

Denote \(z=z(r,a,b)\) the height of the circle \(C_2\) indicating the dependence on the three parameters. Fixing r and b, we have

The function \(b \mapsto b\ \mathrm { arc}\cosh (r/b)\) defined in the interval (0, r) has a unique maximum at a value \(b_0\) which satisfies \(\mathrm {arc}\cosh (r/b_0)=r/\sqrt{r^2-b_0^2}\). The value of \(b_0\ \mathrm { arc}\cosh (r/b_0)\) is \(rb_0/\sqrt{r^2-b_0^2}\) which increases until \(\infty \) as \(r\rightarrow \infty \). Thus we can begin with a value a sufficiently big so z(r, a, b) is bigger than \(z=1\), that is, the height where we want to place the circle \(C_2\). Once obtained the value of a, we decrease the value of b until to get the height of the circle \(C_2\) at \(z=1\).

By the values obtained in Table 1, we conclude the next facts:

1. 1.

   Fixing r, the overlapping distance \({\mathcal {O}}_{1,2}\) decreases with the parameter a.

2. 2.

   The value b that gives the desired height \(z=1\) is a decreasing function on the parameter a.

3. 3.

   Fixing r, there exists a number \(\alpha \) which is the infimum for the values a and the supremum for the values b, where the height \(z=1\) is attained. For this \(\alpha \), the corresponding overlapping distance \({\mathcal {O}}_{1,2}(r)\) is the infimum of all \({\mathcal {O}}_{1,2}(a,b)\) where r and h are keeping fix.

4. 4.

   As r increases, the value \({\mathcal {O}}_{1,2}(r)\) decreases.

By Theorem 1 we know that if

then there is not a (symmetric) Riemann minimal surface spanning \(C_1\cup C_2\). Our calculations cannot compute this quotient for large values of r because Mathematica uses the elliptic integrals of type E and F and the errors appear for large values of r. We have obtained the value of \({\mathcal {O}}_{1,2}\) at least for \(r=50\), where the constants a and b differ up to an accuracy of ten thousandths. See Table 2.

It is an open problem to estimate a sharper upper bound (if there is) for the quotient \({\mathcal {O}}_{1,2}(r)/h\) in (10).

In the last part of this section, we study the estimate for the vertical distance given in (7). It is not expectable a sharp bound for a general situation because inequality (7) holds for any Riemann minimal surface. Again we consider the symmetric case and after a homothety we suppose that the radii of the boundary circles are \(r=2\). Here we fix two circles \(C_1\) and \(C_2\) of radii \(r=2\) situated in each side of \(\varPi \). Let d be the lateral distance which it agrees with the value 2c(r). Now the steps to follow in the calculations have been:

1. 1.

   Fix the lateral distance d, that is, fix the value c(r).

2. 2.

   Take a value b, the waist, where we know that \(b<r\).

3. 3.

   For each value of b, calculate the parameter of a so we have \(c(r)=d/2\).

4. 4.

   Once we know a and b, compute the height of \({\mathcal {R}}_r(a,b)\) which agrees with 2z(r) and compare with the estimate (7).

See Table 3. We have observed that fixed the value of b, the value of c(r) is decreasing on a. Thus, when we fix the value of d, for each value of b, we take \(a=b\) such that \(c(r)>d/2\). Then, we increase the parameter a so c(r) decreases until we get the desired lateral distance 2c(r). Finally we compute the height h of the Riemann minimal surface and we compare the estimate (7) with \(2(r-c(r))/M\).

Remark 1

By the computations obtained in Table 3 we point out that for two given circles \(C_1\cup C_2\) (in this case of radius \(r=2\)) separated a fixed vertical distance h, there exist two Riemann minimal surfaces spanning \(C_1\cup C_2\). Indeed, in each one of the cases \(d=1\), \(d=2\) and \(d=3\), we observe that there exists a maximum of the vertical distance \(h_{max}\) (for \(d=1\), \(d=2\) and \(d=3\), the value \(h_{max}\) it is approximately 2.50766, 2.16727 and 1.48700, respectively). For values of h close to \(h_{max}\) with \(h<h_{max}\) we have two pairs \((a_1,b_1)\) and \((a_2,b_2)\) such that \({\mathcal {R}}_r(a_1,b_1)\) and \({\mathcal {R}}_r(a_2,b_2)\) have the same boundary. This is expectable by the results of Meeks and White in [5] that assert under certain configurations that given two curves lying in parallel planes, these curves are the boundary of exactly two minimal annuli, one stable and one unstable. As in the case of the catenoid, it is expectable that the Riemann minimal surface with least waist is unstable while the Riemann minimal surface with biggest waist is stable.