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Crista egregia: a geometrical model of the crista ampullaris, a sensory surface that detects head rotations

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Abstract

The crista ampullaris is the epithelium at the end of the semicircular canals in the inner ear of vertebrates, which contains the sensory cells involved in the transduction of the rotational head movements into neuronal activity. The crista surface has the form of a saddle, or a pair of saddles separated by a crux, depending on the species and the canal considered. In birds, it was described as a catenoid by Landolt et al. (J Comp Neurol 159(2):257–287, doi:10.1002/cne.9015902071972). In the present work, we establish that this particular form results from principles of invariance maximization and energy minimization. The formulation of the invariance principle was inspired by Takumida (Biol Sci Space 15(4):356–358, 2001). More precisely, we suppose that in functional conditions, the equations of linear elasticity are valid, and we assume that in a certain domain of the cupula, in proximity of the crista surface, (1) the stress tensor of the deformed cupula is invariant under the gradient of the pressure, (2) the dissipation of energy is minimum. Then, we deduce that in this domain the crista surface is a minimal surface and that it must be either a planar, or helicoidal Scherk surface, or a piece of catenoid, which is the unique minimal surface of revolution. If we add the hypothesis that the direction of invariance of the stress tensor is unique and that a bilateral symmetry of the crista exists, only the catenoid subsists. This finding has important consequences for further functional modeling of the role of the vestibular system in head motion detection and spatial orientation.

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Acknowledgments

This study was founded by the European Project CLONS (Closed-loop vestibular neural prosthesis; FP7-ICT-2007.8.0 FET Open, 225929). We would like to thank Prof. Silvestro Micera for his support. Moreover, we are deeply thankful to three reviewers for their constructive comments, which helped us to greatly improve the manuscript.

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Correspondence to Prisca Marianelli.

Appendices

Appendix A: Further insights into crista and hair cell spatial organization

1.1 Shape of the crista ampullaris

The global shape of the cristae varies with the species. It was described for anamniotes (fish and amphibians), and for amniotes (reptiles, birds, mammals).

In fish, the crista of all SCCs is saddle-shaped and narrow at the center. For vertical SCCs, the crista is broader at the two components of the boundary (named planum semilunatum). In frogs, the cristae of the vertical SCCs have a dumbbell shape, with a prominent arched surface in its middle, but the crista of the lateral canal has a single club-shaped receptor zone located at one end. From this enlarged receptor area, a single cupular mass projects along the wall to reach the apex, and a somewhat thinner portion arises from the remainder of the crista receptor area and completes the major part of the cupula (Hillman 1974).

In amniotes, for instance in skates, lizards, turtles, pigeons, most rodents (mouse, rats, gerbil but not chinchilla), cats, monkeys and humans, the cristae of the vertical canals (posterior and anterior) can have the shape of a dumbbell. They present two hemi-cristae, each one with a central zone, separated by a septum cruciatum (Fig. 2; Lindeman 1969). In this case, the cupula extends from the receptor base as two arching masses connected across the center by a thinner region. In contrast, the horizontal canal possesses in general only one central area. Morphological variations between horizontal and vertical SCCs are not surprising; horizontal canals appeared more recently in evolution, and use different gene expressions for their development. For the lateral SCC, depolarization of HCs occurs when the force is exerted toward the utriculus, but for vertical canals, on the contrary, depolarization occurs when the force is exerted toward the duct (Fig. 2).

Morphological studies performed by Landolt et al. (1972) on pigeons, highlighted that the surface geometry of the cristae of the anterior and posterior membranous ampullae is significantly different from that of the lateral membranous ampulla. When viewed from the top, the contents of the ampullae associated with the vertical ducts are reminiscent of a cross. The shorter arms of this cross consist of two protuberances, the eminentiae cruciatae, and the umbo between them, the torus septi. Thus, there are morphologically ‘two cristae’ within each ampulla. Each of these ‘two cristae’ is saddle-shaped, but the crista ampullaris of the lateral membranous ampulla is a v-shaped ridge that traverses the short axis of the lumen of the membranous ampulla, like a catenoid. However, in the three ampullae, the crista ampullaris exhibits a bilateral symmetry about a medial plane to the torus, and another bilateral symmetry about the medial transversal plane that is parallel to the cupula (Fig. 2, Panel c).

The crista ampullaris reaches one-third of the ampullary height and contains the neuroepithelium, the blood vessels and the nerve fibers. The neuroepithelium is mostly made up of hair cells and of supporting cells that are in great part responsible for the biomechanical properties of the crista and of the rheological properties of the gel in the cupula; the cells adjacent to the sensory epithelium also contribute to the structure of the gel (Lindeman 1969; Dohlman 1971).

1.2 The hair cells on the crista

In amniotes, at least two different types of hair cells (HCs) exist: the type I cells are enveloped by calyces dendrites of afferent neurons, on the inner side of the crista epithelium, but the type II cells are contacted by bouton terminations (Wersäll 1954, 1956). (Note the majority of afferent neurons in mammals are dimorphic, contacting both type I and type II HCs.) However, the main physiological gradient is given by the position in the epithelium, for instance on crista, most phasic cells are in the center and more tonic ones are in the periphery (Goldberg and Brichta 1998; Eatock and Songer 2011). In birds, type I HCs are restricted to the center of the crista, but in mammals, type I HCs are found everywhere, and in primates, there is a dominance of type I cells on the whole crista (cf. Lysakowski and Goldberg 2008 on squirrel monkey). However, the crista of mammals has a concentric organization with three types of afferent neurons and HC physiology and morphology: a center, an intermediary zone and a periphery.

The hair cell density is not uniform over the crista surface; it is significantly higher at the periphery than at the center (Hillman 1974; Lindeman 1969). The density difference between the center and periphery is more marked for type I cells, while type II cells were distributed more uniformly between the summit and the periphery of the cristae (Goldberg and Brichta 1998). The hair bundles are shorter in the central region than in the peripheral regions of the cristae (cf. also Njeugna et al. 1996).

The hair bundles (HBs) of the HCs have a variety of morphologies and physiologies for all the epithelia of vestibular end organs, canal’s crista or otolith’s macula (Brichta et al. 2002).

The stereocilia of one HB are inserted at the apex of the HC in regular arrangement, and they are interconnected in at least three ways: basal connections, lateral contact and tip links, (Howard et al. 1988). It was proved (at least for vibratory motions) that the bundle deflection is synchronous and coherent (Kozlov et al. 2006).

Several studies have reported the insertion of the kinocilium in the cupula shell, and filaments forming veils and tubes, connecting the stereocilia to the subcupula space (Suzuki et al. 1984; Rüsch and Thurm 1989).

1.3 Cupula structure

The cupula was first described with precision by Steinhausen (1933) as a gelatinous substance extending to the roof of the ampulla and out to its lateral walls.

Different theories were proposed for the cupula. Zalin (1967) proposed that the cupula is suspended from the ampulla roof. Dohlman (1971) proposed that the cupula is moving like a piston, and suggested that the endolymph fluid circulates at the roof of the ampulla, not along the crista, but later (1977, 1980) he suggested a different view that endolymph circulates in the subcupular space. Rüsch and Thurm (1989) gave arguments favouring a different composition of the fluid in the cupula antrum (Muller 1999).

In toadfish, Yamauchi et al. (2001) have shown that the cupula can disconnect from the roof, but stays attached on the boundary of the crista.

In addition, Dohlman (1977, 1980) proposed that rigid structures, like filaments or tubes, are linking the stereocilia to the cupula. Such a filamentous structure was also observed by Suzuki et al. (1984), and Rüsch and Thurm (1989), Takumida (2001). However, this structure could depend on species differences. For instance Silver et al. (1998) reported the absence of such structures in toadfish. The optimality of a thin subcupular space for augmenting sensibility was proved by Muller (1999).

From the work of Hillman and McLaren (1979), studying frogs, and Silver et al. (1998), studying oyster toadfish (opsanus tau), we can infer that the cupula complex is made up of four parts (Fig. 3): lateral wings on both sides, joint central pillars in the middle (forming the cupular shell), and a cupula antrum near the crista, filled with isotropic gel, reinforced with collagen running vertically (cf. Hunter-Duvar and Hinojosa 1984 for mammals). The cupular shell is made of connective tissue fibers, densely packed and cross-linked. Accordingly, Takumida (2001) distinguished between cupula in the narrow sense (for cupular shell) and subcupular meshwork (for the cupular antrum).

The cupular antrum contains an aqueous media into which the hair cell bundles (formed by multiple stereocilia and a single kinocilium) project, which arise from the apical surface of the sensory hair cells (Hillman and McLaren 1979). This gel can be free of material, as in toadfish (Silver et al. 1998), but it cans also be full of connecting structures, as in guinea pig (Takumida 2001). In guinea pigs, Wersäll (1956) described fine vertical canals where HC bundles are located. This ensemble of connective thin fibers and ciliae has been called by Takumida (2001) the subcupular meshwork. This is a mass of extra cellular, highly specialized material (Landolt et al. 1972; Silver et al. 1998).

Takumida proposed a functional relationship between the sensory hair cells and the cupula (Takumida 2001). According to his observation, the highly cross-linked isotropic texture of the cupular shell indicates that this layer could function as a rigid plate and equally distributes the forces of inertia by the large bulk of the whole cupula to all sensory hair bundles. Shear stress due to relative acceleration of the cupula results in shear strain of the cupula itself. The energy coming from the cupula motion is transmitted to the sensory hairs directly or indirectly through the subcupular meshwork. This observation of Takumida is the main basis of our hypothesis of invariance. In fact, the function of the cupular system is to transform the three-dimensional cupula deformation into a one dimensional hair cell bending, and we interpret the hypothesis of equal distribution of forces in Takumida as a hypothesis of dimensional reduction by a one-parameter group of symmetry.

1.4 Cupula models

There is now experimental evidence that the cupula displaces like a diaphragm when subject to a pressure gradient (Hillman and McLaren 1979; Yamauchi et al. 2001).

The standard theory is that the cupula is analogous to a drum membrane (Landolt et al. 1972). Also Van Buskirk (1976) modeled the cupula as an elastic membrane that spans the whole ampulla in the form of a diaphragm. (For a review of the cupula models, see also McLaren and Hillman 1979; Hillman and McLaren 1979.)

According to the experiments performed on bullfrog cupula, McLaren and Hillman (1979) and Hillman and McLaren (1979) suggested that the base of the cupula slides over the surface of the crista and shears the subcupular space between the crista and the cupula. Note that the shear strain across the entire transverse width of the crista is either excitatory or inhibitory to all hair cells at the same time (Rabbitt et al. 2001), depending on the transverse displacement of the cupula wings.

Kondrachuk et al. (1987) gave a mathematical model of the time-course variations of the cupula; in particular, they discussed the viscosity–elasticity properties of the cupula for the applicability of a membrane model. Others (e.g., Astakhova 1989, 1990) have considered the system as being composed of an elastic piston (cupula) in a viscous fluid (endolymph).

Vega et al. (2008) performed a comparison between the two models, which demonstrated that cupular behavior dynamics under periodic stimulation is equivalent for both the piston and the membrane cupular models.

Therefore, although the maximum cupular displacement occurs near the geometric center of the ampulla, the maximum shear strain occurs precisely at the level where the sensory hair bundles project into the cupula. This point was underlined in the work of (Muller 1999). Because it is the relative displacement between adjacent cilia that ultimately leads to transduction channel gating, the shear strain in the cupula appears to be ideally suited to activate hair cells. For vibratory rotations of the head, the coupled dynamics of the fluid in the canals and of the cupula imply the existence of a critical frequency band where the transverse cupula displacement is proportional to, and in phase with, the angular velocity of the head (Wilson and Melvill Jones 1979; Rabbitt et al. 2001). In this band, the shear strain is proportional to the volume displacement at the level of the cupula. Highstein et al. (2005) and Rabbitt et al. (2001) have made a simulation of that point.

There is experimental evidence that the cupula (i) has a certain stiffness and resists deformation from its resting position, (ii) has viscosity and resists the rate of change of deformation from the resting position, and (iii) has a spatially non-uniform deflection field (Damiano and Rabbitt 1996). The rotational stiffness \(K\) in \(SI\) units is \(K = 7.1 \cdot 10^{-11}\) N m/rad (Grant and Van Buskirk 1976). The cupula completely spans the cross section of the ampulla. It is a relative thick structure (one-third of the height, for instance 0.15 mm width for 0.45 mm height in the axolotl, Vega et al. 2008), but it is assumed that it deforms uniformly through the thickness, a restrictive assumption also found in Damiano and Rabbitt (1996). This conforms to the functional plan suggested by Takumida (2001).

In our study, we have modeled the cupula system as an inhomogeneous Newtonian visco-elastic solid, composed of four homogeneous domains: the wings (or two side walls), the cupula proper (or two cupula pillars), the cupula antrum and the crista. The incompressibility of the gel in the antrum is affirmed by Selva et al. (2009); Yamauchi et al. (2001); Kassemi et al. (2005).

Most of the present study concerns the antrum, viewed as a hydrogel (Selva et al. 2009). Thus, we do not assume homogeneity of the cupula, but we assume homogeneity in the bottom part of the cupula (antrum) in the vicinity of the crista. This hypothesis is explicitly made by most of the model papers (Selva et al. 2009; Vega et al. 2008; Damiano 1999). As the displacement in the cupula is a question of a few micrometers, frequently a dozen nanometers, our model equations will correspond to the linear Hooke law.

Appendix B: Analysis and demonstrations

1.1 Elasticity equations

The stress tensor \(\sigma ^{ij}\) in the solid is the generator of forces: along an infinitesimal piece of oriented surface described by a co-vector \(\alpha _j,j=1,2,3\) the applied force is given by

$$\begin{aligned} f^{i}=\sum _j \sigma ^{ij}\alpha _j. \end{aligned}$$
(60)

The tensor \(\sigma ^{ij}\) is symmetric (i.e., \(\sigma ^{ij}=\sigma ^{ji}\) for any pair \(i,j\) of indices).

We make the hypothesis that the cupula antrum has a linear visco-elastic model, which is incompressible.

The assumption of linear elasticity (Hooke law) is that \(\sigma ^{ij}\) depends linearly of the strain tensor \(u^{ij}\), which is induced by the displacement:

$$\begin{aligned} \sigma ^{ij}(x,t)=2\mu (x)u^{ij}(x,t)+\lambda (x)Tr(u)g^{ij}(x,t), \end{aligned}$$
(61)

where \(\mu \) and \(\lambda \) are the Lamé coefficients, respectively shear modulus and compression modulus, \(g^{ij}\) the ambient (contravariant) metric tensor and \(Tr(u)\) the trace of strains tensor:

$$\begin{aligned} Tr(u)(x,t)=\sum _{ij}g_{ij}(x)u^{ij}(x,t), \end{aligned}$$
(62)

where \(g_{ij}\) and \(g^{ij}\) are inverse matrices. The term \(Tr(u)\) commands the local variation of volume during the deformation. We assume in what follows that \(Tr(u)=0\), which means that the viscoelastic solids and gels in presence are incompressible. This is compatible with the view that the continuous constituent of the gel looks like water.

The coefficient \(\mu \) is supposed constant in each of the three compartments and varying abruptly at the frontiers, being much smaller in the antrum than in the other parts. We will work in the linear approximation of elasticity, where the deformation tensor \(u^{ij}\) is supposed linear in the displacement vector field \(u^{i}\). In fact \(u^{ij}\) is given by a symmetrization of the covariant derivative of \(u^{i}\):

$$\begin{aligned} (\nabla _ju)^{i}=\partial u^{i}/\partial x^{j}+\sum _k\varGamma _{jk}^{i}u^{l}. \end{aligned}$$
(63)

On the sensory epithelium the boundary condition is an expression of the equilibrium of forces at the frontier between each compartments, that is between the cupula fibers and cupula antrum and between the cupula antrum and the crista:

$$\begin{aligned} \sum _i\sigma ^{0}_{ij}n^{i}+\sum _i\sigma ^{1}_{ij}n^{i}=0 \end{aligned}$$
(64)

where \(n=\sum _in^{i}\partial _i\) denotes the unit normal at the frontier, and the indices \(0,1\) denote the two media around the frontier. (The covariant tensors \(\sigma ^{0}_{ij}, \sigma ^{1}_{ij}\) are obtained by rising indices with the matrix \(g_{kl}\).)

The dynamic is described by the Newton law, equating the divergence of the tensor \(\sigma ^{ij}\) with the sum of the applied forces and the covariant acceleration of the displacement \(u^i\), which is the inertial force. This is expressed by the Eqs. (17) and (19) at equilibrium positions, which is attained at the maximum displacement. Note that the total expense of energy depends on this case.

The other equation is Eq. (18) which expresses the incompressibility.

The stationarity of the direction of the gradient of pressure is an invitation to choose coordinates \(x^1,x^2,x^3\) such that \(\partial _1=\partial /\partial x^1\) is the direction of the pressure gradient, \(\partial _2=\partial /\partial x^2\) is orthogonal to \(\partial _1\) on \(\varSigma \), and \(x^3\) denotes the coordinate along the unit normal vector \(\mathbf{n}^{0}\) of \(\varSigma \) in \(\mathbb {R}^{3}\). The coordinate \(x^3\) will be the crucial parameter for the analysis of tensors order by order.

We will first prove that the Navier equation and the incompressibility imply that the components \(\beta _{i}\) are zero up to order \(3\) in the distance to \(\varSigma \) except for the component \(\beta _{1}\), dual to the gradient of pressure, which must be of order two.

During movements, the difference of pressure exerted on the walls of the cupula moves them, generating variations of the internal energy density inside the antrum:

$$\begin{aligned} \mathrm{d}\mathcal {E}=T\mathrm{d}S +\sum _{ij}\sigma ^{ij}(x,t)\mathrm{d}u_{ij}(x,t), \end{aligned}$$
(65)

where \(T\) denotes the temperature and \(S\) the entropy. The \(u_{ij}\) appears as the momentum of the \(\sigma ^{ij}\). If their traces were nonzero, the second term in the right member would contain a term \(-p\mathrm{d}V\), where \(p\) is the pressure and \(V\) the volume.

The total variation of kinetic energy is given by Eq. (21). From that, we will deduce that the energy dissipation due to viscosity in the cupula is proportional to the integral of the square norm of the tensor \(\sigma \) for the largest displacement. The expression would not be different for a viscoelastic fluid, with a memory effect or with a relaxation effect. The shear forces exerted on the crista epithelium is the component \((\sigma .\mathbf{n})^{i}=\sum _j\sigma ^{ij}n_j\) along the normal vector; the transmission of information through HCs implies that it gives a nonzero contribution in this integral. The computation of the term \(u_{13}\) under our constraint of invariance will show that the total surface of \(\varSigma \) has to be minimal if we want minimize the energy loss.

Remark that similar results should be obtained with the same proof for a Kelvin model of visco-elastic solid, or for a Maxwell model of visco-elastic fluid. Even for nonlinear models the result could be justified, because at equilibrium, for maximal displacement, the system is well approached by a linear one. However, for a nonlinear model, other sources of energy loss have to be taken in account.

1.2 From statics to geometry

We adopt the notations and hypotheses of Sect. 2.

The four equilibrium equations:

For the Hodge operator \(*\), we have

$$\begin{aligned} *\mathrm{d}x^1&=-\sqrt{g}g^{12}\mathrm{d}x^1\wedge \mathrm{d}x^3+\sqrt{g}g^{11}\mathrm{d}x^2\wedge \mathrm{d}x^3,\nonumber \\ *\mathrm{d}x^2&=-\sqrt{g}g^{22}\mathrm{d}x^1\wedge \mathrm{d}x^3+\sqrt{g}g^{12}\mathrm{d}x^2\wedge \mathrm{d}x^3,\nonumber \\ *\mathrm{d}x^3&=\sqrt{g}\mathrm{d}x^1\wedge \mathrm{d}x^2. \end{aligned}$$
(66)

and

$$\begin{aligned} *(\mathrm{d}x^1\wedge \mathrm{d}x^2)&=\sqrt{g}g^{11}g^{22}\mathrm{d}x^3,\nonumber \\ *(\mathrm{d}x^2\wedge \mathrm{d}x^3)&=\sqrt{g}g^{22}\mathrm{d}x^1-\sqrt{g}g^{12}\mathrm{d}x^2,\nonumber \\ *(\mathrm{d}x^1\wedge \mathrm{d}x^3)&=\sqrt{g}g^{12}\mathrm{d}x^1-\sqrt{g}g^{11}\mathrm{d}x^2. \end{aligned}$$
(67)

Then

$$\begin{aligned} *&\mathrm{d}\beta =\sqrt{g}g^{11}g^{22}(\beta ^{2}_1-\beta ^{1}_2)\mathrm{d}x_3 +(\sqrt{g}g^{22}(\beta ^{3}_2-\beta ^{2}_3) + \nonumber \\&+\sqrt{g}g^{12}(\beta ^{3}_1-\beta ^{1}_3))\mathrm{d}x_1 -(\sqrt{g}g^{11}(\beta ^{3}_1-\beta ^{1}_3)+\nonumber \\&+\sqrt{g}g^{12}(\beta ^{3}_2-\beta ^{2}_3))\mathrm{d}x_2. \end{aligned}$$
(68)

This gives:

$$\begin{aligned}&\mathrm{d}*\mathrm{d}\beta =-[\partial _1(\sqrt{g}g^{11}(\beta ^{3}_1-\beta ^{1}_3) +\sqrt{g}g^{12}(\beta ^{3}_2-\beta ^{2}_3))\nonumber \\&\quad +\partial _2((\sqrt{g}g^{22}(\beta ^{3}_2-\beta ^{2}_3) +\sqrt{g}g^{12}(\beta ^{3}_1-\beta ^{1}_3))]\mathrm{d}x_1\wedge \mathrm{d}x_2\nonumber \\&\quad +[\partial _1(\sqrt{g}g^{11}g^{22}(\beta ^{2}_1-\beta ^{1}_2))-\partial _3(\sqrt{g}g^{22}(\beta ^{3}_2-\beta ^{2}_3) \nonumber \\&\quad +\sqrt{g}g^{12}(\beta ^{3}_1-\beta ^{1}_3))]\mathrm{d}x_1\wedge \mathrm{d}x_3\nonumber \\&\quad +[\partial _2(\sqrt{g}g^{11}g^{22}(\beta ^{2}_1-\beta ^{1}_2)-\partial _3(\sqrt{g}g^{11}(\beta ^{3}_1-\beta ^{1}_3)\nonumber \\&\quad +\sqrt{g}g^{12}(\beta ^{3}_2-\beta ^{2}_3))] \mathrm{d}x_2\wedge \mathrm{d}x_3. \end{aligned}$$
(69)

We have chosen the coordinates to guaranty that at maximum deflection, the equation \(\mathrm{d}p=-a\mathrm{d}x^{1}\) holds in the antrum compartment for a certain constant \(a\), which depends on the choice of \(x^1\) in covariant way, thus

$$\begin{aligned} -*\mathrm{d}p=a\sqrt{g}g^{11}\mathrm{d}x_2\mathrm{d}x_3-a\sqrt{g}g^{12}\mathrm{d}x_1\mathrm{d}x_3. \end{aligned}$$
(70)

So the coefficient of \(\mathrm{d}x^1\mathrm{d}x^2\) in \(*\mathrm{d}*\mathrm{d}\beta \) must be zero, which gives the first equation:

$$\begin{aligned}&\partial _1(\sqrt{g}g^{11}(\beta ^{3}_1-\beta ^{1}_3) +\sqrt{g}g^{12}(\beta ^{3}_2-\beta ^{2}_3))\nonumber \\&\quad +\partial _2((\sqrt{g}g^{22}(\beta ^{3}_2-\beta ^{2}_3) +\sqrt{g}g^{12}(\beta ^{3}_1-\beta ^{1}_3))=0 \end{aligned}$$
(71)

For degree one forms in dimension three, we have \(**=Id\), thus the coefficient of \(\mathrm{d}x_1\mathrm{d}x_3\) gives the second equation:

$$\begin{aligned}&\partial _1(\sqrt{g}g^{11}g^{22}(\beta ^{2}_1-\beta ^{1}_2)) -\partial _3(\sqrt{g}g^{22}(\beta ^{3}_2-\beta ^{2}_3) \nonumber \\&\quad +\sqrt{g}g^{12}(\beta ^{3}_1-\beta ^{1}_3))=-\frac{a}{\mu }\sqrt{g}g^{12}. \end{aligned}$$
(72)

And the coefficient of \(\mathrm{d}x_2\mathrm{d}x_3\) gives the third equation:

$$\begin{aligned}&\partial _2(\sqrt{g}g^{11}g^{22}(\beta ^{2}_1-\beta ^{1}_2)) -\partial _3(\sqrt{g}g^{11}(\beta ^{3}_1-\beta ^{1}_3) \nonumber \\&\quad +\sqrt{g}g^{12}(\beta ^{3}_2-\beta ^{2}_3))=\frac{a}{\mu }\sqrt{g}g^{11} \end{aligned}$$
(73)

We get one more equations by expressing that \(\mathrm{d}*\beta =0\). From the following formula:

$$\begin{aligned} *\beta&= (\sqrt{g}g^{11}\beta ^{1}+\sqrt{g}g^{12}\beta ^{2})\mathrm{d}x_2\mathrm{d}x_3\nonumber \\&-(\sqrt{g}g^{12}\beta ^{1}+\sqrt{g}g^{22}\beta ^{2})\mathrm{d}x_1\mathrm{d}x_3 +\sqrt{g}\beta ^{3}\mathrm{d}x_1\mathrm{d}x_2,\nonumber \\ \end{aligned}$$
(74)

the fourth equilibrium equation is:

$$\begin{aligned}&\partial _3(\sqrt{g}\beta ^{3})+\partial _1(\sqrt{g}g^{11}\beta ^{1} +\sqrt{g}g^{12}\beta ^{2})\nonumber \\&\quad +\partial _2(\sqrt{g}g^{12}\beta ^{1}+\sqrt{g}g^{22}\beta ^{2})=0 \end{aligned}$$
(75)

Proof of Proposition 1

Along the surface \(\varSigma \), that is for \(x_3=0\), we know from the boundary conditions that \(\beta =0\), and that \(\beta _3^{1}=\beta _3^{2}=\beta _3^{3}=0\). And our choice of coordinates \(x_1,x_2\) makes that \(g_{12}=0\), thus also \(g^{12}=0\). Thus, from the third equation (Eq. 73), we obtain \(\beta ^{1}_{33}=a/\mu \); this is because all other terms are of order two in \(x_3\). From the fourth equation (Eq. 75), we get \(\beta ^{3}_{33}=0\); because it would give the only possible term of order one in \(x_3\). Then the second equation (Eq. 72), gives \(\beta ^{2}_{33}=0\), because it is the only possible term of order zero in \(x_3\) in it. Then, the first equation (Eq. 71), by looking at terms of order one in \(x_3\), implies that along \(\varSigma \) we have \((\sqrt{g}g^{11})_1=0\), which is equivalent to \((g^{11}/g^{22})_1=0\) or to \((g_{11}/g_{22})_1=0\), because \(g=g_{11}g_{22}=1/g^{11}g^{22}\). (At first sight it seems strange that we get constraints on the metric, but this is not a constraint on \(\varSigma \), this is a constraint on the coordinates \(x_1,x_2\), resulting from their compatibility with \(\mathrm{d}p\).)

When looking again at the fourth equation (Eq. 75), the only term of order two must \(3\sqrt{g}\beta _{333}^{3}x_3^{2}\), that is due to \((\sqrt{g}g^{11})_1=0\) and to the fact that \(a_1=0\), thus we get \(\beta _{333}^{3}=0\).

Thus, we have obtained from the equilibrium equations the identities along \(\varSigma \) which are written in Proposition 1. \(\square \)

Proof of Proposition 3

We write the Eq. (73) at the order one in \(x_3\). The coefficients \(g^{ij}\) are obtained by inversion of the matrix \(g_{ij}\):

$$\begin{aligned}&g^{11}\approx (g_{11}^{0})^{-1}-2g^{-1}x_3b_{22}\nonumber \\&g^{22}\approx (g_{22}^{0})^{-1}-2g^{-1}x_3b_{11}\nonumber \\&g^{12}=g^{21}\approx 2g^{-1}x_3b_{12}. \end{aligned}$$
(76)

Thus, at the first order in \(x_3\), we get

$$\begin{aligned} \sqrt{g}g^{11}\approx \sqrt{\frac{g_{22}^{0}}{g_{11}^{0}}}(1-2x_3b_2^{2}). \end{aligned}$$
(77)

Now we write \(b\) for the third coefficient in \(x_3\) of \(\beta ^{1}\), the Eq. (73) at order one, can be written

$$\begin{aligned} \frac{a}{\mu }(1-2b_2^{2}x_3)&\approx \partial _3((1-2b_2^{2}x_3+\cdots ) \left( \frac{a}{\mu }x_3+3bx_3^{2})\right) ,\nonumber \\&\approx \frac{a}{\mu }+6bx_3-4\frac{a}{\mu }b_2^{2}x_3+\cdots \end{aligned}$$
(78)

which gives the result on \(\beta \), when using \(b_1^{1}+b_2^{2}=0\). But \(u_1=g^{11}\beta ^{1}+g^{12}\beta ^{2}\) thus at the order \(3\) in \(x_3\) we have:

$$\begin{aligned} u_1&=g^{11}\beta ^{1}\approx (g_{11}^{0})^{-1}(1-2b_2^{2}x_3 +O(x_3^{2}))\nonumber \\&\quad \left( \frac{a}{2\mu }x_3^{2}+\frac{1}{3}\frac{a}{\mu }b_2^{2}x_3^{3} +O(x_3^{4})\right) ,\nonumber \\&\approx (g_{11}^{0})^{-1}\left( \frac{a}{2\mu }x_3^{2} -\frac{2}{3}\frac{a}{\mu }b_{2}^{2}x_3^{3}+O(x_3^{4})\right) . \end{aligned}$$
(79)

which gives the second result.\(\square \)

Proof of Proposition 4

We use that at this order, the only derivative \(\partial _ku_l\) that is nonzero is \(\partial _3u_1\) we just have computed and that the only terms \(\varGamma _{lm}^{k}u_m\) that are nonzero are the ones where \(m=1\), plus the fact that the only \(g^{lm}\), which are nonzero on \(\varSigma \) are the diagonal ones, i.e., for \(l=m\). Then, we have to compute the symbols \(\varGamma _{11;k},\varGamma _{21;k},\varGamma _{31;k}\); at the order zero, they are given by the following formulas:

$$\begin{aligned}&\varGamma _{11;1}=\frac{1}{2}\partial _1g_{11},\quad \varGamma _{11;2}=-\frac{1}{2}\partial _2g_{11},\quad \varGamma _{11;3}=b_{11};\nonumber \\&\varGamma _{21;1}=\frac{1}{2}\partial _2g_{11},\quad \varGamma _{21;2}=-\frac{1}{2}\partial _1g_{22},\quad \varGamma _{21;3}=b_{12};\nonumber \\&\varGamma _{31;1}=-b_{11},\quad \varGamma _{31;2}=-b_{12},\quad \varGamma _{31;3}=0. \end{aligned}$$
(80)

The proposition follows from the definitions by grouping the terms.

For instance:

$$\begin{aligned} u_{11}=g^{11}u_1^{1}\approx g^{11}\varGamma _{11}^{1}u_1\approx (g^{11})^{2}\varGamma _{11;1}u_1; \end{aligned}$$
(81)

the same for \(u_{22}\) and \(u_{33}\). Now for the mixed symbols:

$$\begin{aligned} u_{21}&=\frac{1}{2}(g^{11}u_2^{1}+g^{22}u_1^{2})\approx \frac{1}{2}(g^{11}\varGamma _{11}^{2}u_1+g^{22}\varGamma _{21}^{1}u_1) \nonumber \\&\approx \frac{u_1}{2}g^{11}g^{22}(\varGamma _{11;2}+\varGamma _{21;1})\approx 0.\nonumber \\&u_{23}=\frac{1}{2}(g^{22}u_3^{2}+g^{33}u_2^{3})\approx \frac{1}{2}(g^{22}\varGamma _{21}^{3}u_1+g^{33}\varGamma _{31}^{2}u_1) \nonumber \\&\approx \frac{u_1}{2}g^{22}g^{33}(\varGamma _{21;3}+\varGamma _{31;2})\approx 0. \end{aligned}$$
(82)

And for the last one:

$$\begin{aligned} \widetilde{u}_{13}&\approx g^{11}u_3^{1}\approx g^{11}\varGamma _{11}^{3}u_1\approx g^{11}\varGamma _{11;3}u_1 \nonumber \\&\approx b_{11}(g_{11}^{0})^{-2}(\frac{a}{2\mu }x_3^{2}+\cdots );\nonumber \\ \widetilde{u}_{31}&\approx g^{33}u_1^{3}\approx \partial _3u_1+\varGamma _{31}^{1}u_1 \nonumber \\&\approx (g_{11}^{0})^{-1}\left[ \left( \frac{a}{\mu }x_3 +2\frac{a}{\mu }b_1^{1}x_3^{2}+\cdots \right) \right. \nonumber \\&\qquad \qquad \qquad \left. +\varGamma _{31}^{1}\left( \frac{a}{2\mu }x_3^{2} +O(x_3^{3})\right) \right] ; \end{aligned}$$
(83)

that is

$$\begin{aligned} \widetilde{u}_{31}&\approx \frac{a}{g_{11}^{0}}\left( x_3+2\frac{1}{g_{11}^{0}}b_{11}x_3^{2} +\frac{1}{2}\varGamma _{31;1}x_3^{2}\right) \nonumber \\&\approx \frac{a}{g_{11}^{0}}\left( x_3+2b_1^{1}x_3^{2}-\frac{1}{2}b_1^{1}\right) x_3^{2} \end{aligned}$$
(84)

From here the announced result follows. \(\square \)

Proof of the first corollary of Proposition 4

We have by hypothesis that the trace of \(u_{ij}\) is zero, that is \(u_1^{1}+u_2^{2}=0\), which gives at order zero in \(x_3\):

$$\begin{aligned} g^{11}\partial _1(g_{11})=0, \end{aligned}$$
(85)

but we already had

$$\begin{aligned} g^{11}\partial _1(g_{11})-g^{22}\partial _1(g_{22})=0, \end{aligned}$$
(86)

from the Proposition 1. This is the last equation \(\partial _1(g_{11}/g_{22}) =0\); it gives \(g_22\partial _1(g_{11}-g_{11})\partial _1g_{22}=0\), which implies what we want by definition of \(g^{11},g^{22}\). \(\square \)

Proof of the second corollary of Proposition 4

This follows from the hypothesis that \(u^{13}\) is invariant by \(\partial _1\). \(\square \)

Proof of Proposition 5

This follows from the Eq. (72). Under the hypothesis, the development of the discriminant \(g=g_{11}g_{22}-g_{12}^{2}\) has no term in \(x_3\), and \(g^{12}\approx -g_{12}/g\), then

$$\begin{aligned} \sqrt{g}g^{12}\approx 2g^{-1/2}b_{12}x_3. \end{aligned}$$
(87)

Remark that minimality implies also that \(n^{0}_1.n^{0}_2=0\), thus \(g_{12}\) has no component of order two.

Consequently, at order one in \(x_3\), (Eq. 72) gives

$$\begin{aligned} -2ag^{-1/2}b_{12}x_3&\approx \partial _3(\sqrt{g}g^{22}\partial _3\beta ^{2})\nonumber \\&\approx \sqrt{g_{11}/g_{22}}6cx_3, \end{aligned}$$
(88)

where \(c\) denotes the coefficient of \(x_3^{3}\) in \(\beta ^{2}\). The result follows. \(\square \)

Proof of Proposition 6

We compute \(u_{11}\) at the order three in \(x_3\). We begin by the observation that at order three \(u^{11}\approx g^{11}u_1^{1}\), thus, the only terms which could influence the development are \(\partial _1u_1\), \(\varGamma _{11}^{1}u_1\) and \(\varGamma _{12}^{1}u_2\); but at the order one we have:

$$\begin{aligned} \varGamma _{11;1}&= \frac{1}{2}\partial _1g_{11}\approx -\partial _1 b_{11} x_3,\nonumber \\ \varGamma _{11;2}&\approx -\frac{1}{2}\partial _2g^{0}_{11}-2x_3\partial _1b_{12}+x_3\partial _2 b_{11},\nonumber \\ \varGamma _{12;1}&=\frac{1}{2}\partial _2g_{11}\approx \frac{1}{2}\partial _2g^{0}_{11}-x_3\partial _2b_{11}; \end{aligned}$$
(89)

thus

$$\begin{aligned} \varGamma _{11}^{1}u_1&\approx g^{11}\varGamma _{11;1}u_1+g^{12}\varGamma _{11;2}u_1,\nonumber \\&\approx -g^{11}(x_3\partial _1 b_{11}+x_3b_2^{1}\partial _2g_{11}) \frac{1}{2g^{0}_{11}}\frac{a}{\mu }x_3^{2},\nonumber \\ \varGamma _{12}^{1}u_2&\approx g^{11}\varGamma _{12;1}u_2,\nonumber \\&\approx -g^{11}g^{22}\frac{1}{2}\partial _2g_{11}\frac{1}{3} \frac{a}{\mu }b_2^{1}x_3^{3}. \end{aligned}$$
(90)

We know from Proposition 3 that

$$\begin{aligned} \partial _1u_1\approx \frac{2}{3}\frac{a}{\mu g_{11}^{0}}\partial _1b_1^{1}x_3^{3}. \end{aligned}$$
(91)

However, from the second corollary of the Proposition 4, we have \(\partial _1b_{11}=0\), then:

$$\begin{aligned} u_{11}\approx -(g^{11})^{2}\partial _2g_{11}\frac{a}{2\mu }b_2^{1} \left( g^{11}+\frac{1}{3}g^{22}\right) x_3^{3}. \end{aligned}$$
(92)

Thus, if \(\partial _2g^{0}_{11}\ne 0\), the invariance of \(u_{11}\) implies the invariance of \(b_2^{1}\), which implies the invariance of \(b_{12}\), so the invariance of all the \(b_{ij}\). But if \(g^{0}_{11}\) is constant, we choose the coordinate \(x_2\) such that \(g_{22}\) is constant too, the metric is flat, and we know that a flat minimal surface is a plane, thus in this case too, we get constant \(b_{ij}\). \(\square \)

Appendix C: Geometry. Invariant minimal surfaces

Our aim was to prove the two purely geometrical statements used for characterize the form of the crista surface from its invariance and its minimality.

Proof of Proposition A

By integrating \(\sqrt{g_{22}}\), we can suppose that

$$\begin{aligned} g_{11}=g_1^{2}(x_2),\quad g_{12}=g_{21}=0,\quad g_{22}=1. \end{aligned}$$
(93)

Thus the vectors \(e_1=r_1/g_1,e_2=r_2,e_3=n\), where \(n=e_1\times e_2\), constitute an orthonormal frame. We recall that we denote partial derivatives by indices. By differentiating the scalar products of these vectors we obtain:

$$\begin{aligned} e_{1;1} \cdot e_1&=0,\quad e_{1;1} \cdot e_2+ e_{2;1} \cdot e_1=0, \nonumber \\ e_{2;1} \cdot e_2&=0,\quad e_{3;1} \cdot e_3=0,\nonumber \\ e_{1;2} \cdot e_1&=0,\quad e_{1;2} \cdot e_2+ e_{2;2} \cdot e_1=0,\nonumber \\ e_{2;2} \cdot e_2&=0,\quad e_{3;2} \cdot e_3=0. \end{aligned}$$
(94)

We verify directly that

$$\begin{aligned} e_{1;1} \cdot e_2&=-\frac{1}{g_1}\varGamma _{11,2}=-\frac{1}{2g_1}\partial _2g_{11} =-\partial _2g_1; \nonumber \\ e_{1;2} \cdot e_2&=\varGamma _{12,2}=0; \end{aligned}$$
(95)

thus

$$\begin{aligned} e_{2;1} \cdot e_1=\partial _2g_1,\quad e_{2;2} \cdot e_1=0. \end{aligned}$$
(96)

Moreover, the invariance of \(g_{ij}\) implies

$$\begin{aligned} e_{11}=\frac{r_{11}}{g_1},\quad e_{1;2}=\frac{r_{12}}{g_1}-\frac{g_{1;2}r_1}{g_{11}}; \end{aligned}$$
(97)

which gives by definition of the symbols \(b_{ij}\):

$$\begin{aligned} e_{1;1} \cdot e_3&=b_{11}/g_1,\quad e_{2;1} \cdot e_3=b_{21},\nonumber \\ e_{3;1} \cdot e_1&=-b_{11}/g_1,\quad e_{3;2} \cdot e_2=-b_{12},\nonumber \\ e_{1;2} \cdot e_3&=b_{12}/g_1,\quad e_{2;2} \cdot e_3=b_{22},\nonumber \\ e_{3;2} \cdot e_1&=-b_{21}/g_1,\quad e_{3;2} \cdot e_2=-b_{22}. \end{aligned}$$
(98)

Let us introduce the two \(3\times 3\) matrices \(A_1,A_2\), whose coefficients are respectively the numbers \(A_{1,ij}=e_{j;1} \cdot e_i\) and \(A_{2,ij}=e_{j;2} \cdot e_i\). We get

$$\begin{aligned}&A_1= \begin{pmatrix} 0&{}\partial _2g_1&{}-b_{11}/g_1\\ -\partial _2g_1&{}0&{}-b_{12} b_{11}/g_1&{}b_{12}&{}0 \end{pmatrix},\nonumber \\&A_2= \begin{pmatrix} 0&{}0&{}-b_{12}/g_{1}\\ 0&{}0&{}-b_{22} b_{12}/g_1&{}b_{22}&{}0 \end{pmatrix} \end{aligned}$$
(99)

Consider the variable orthogonal matrix \(Y\) whose columns are \(e_1,e_2,e_3\). We verify easily the following matrix equations for the moving frame \(e_1(x_1,x_2),e_2(x_1,x_2),e_3(x_1,x_2)\):

$$\begin{aligned} \partial _1Y=YA_1,\quad \partial _2 Y=YA_2. \end{aligned}$$
(100)

Our invariance hypothesis implies that \(A_1\) and \(A_2\) are only functions of \(x_2\), not of \(x_1\). Thus, the first of these equations, for each given value of \(x_2\), has constant coefficients, thus it can be solved explicitly by the exponential of matrices:

$$\begin{aligned} Y(x_1,x_2)=Y(a_1,x_2)e^{(x_1-a_1)A_1}. \end{aligned}$$
(101)

And, as \(A_1\) is an antisymmetric matrix, the exponential of \(tA_1\) for each \(t\), is a rotation around a fixed axis, which a priori can depend on \(x_2\), with an angular velocity which a priori can also depend on \(x_2\).

The axis and the angular velocity are given by the following vector:

$$\begin{aligned} v=-b_{12}e_1+\frac{b_{11}}{g_1}e_2+(\partial _2g_1)e_3. \end{aligned}$$
(102)

Observe there is a case where the axis \(v\) and the velocity are not defined, it is when \(b_{11}=b_{12}=\partial _2g_1=0\), i.e., when \(A_1=0\). In this case, \(g_{ij}\) is constant. Then, the surface \(\varSigma \) is flat, and \(\partial _1\) generates a translation, thus \(\varSigma \) is a cylinder, over a plane curve of curvature \(b_{22}(x_2)\).

In all the other cases let us verify that the angular velocity vector \(v\) is independent of \(x_2\).

If we compute the derivative of \(v\) by \(\partial _1\), we find

$$\begin{aligned} \partial _2v&=\left( -\partial _2b_{12}\!-\!\frac{b_{12}}{g_1}\partial _2g_1\right) e_1\!+\! \left( \partial _2\left( \frac{b_{11}}{g_1}\quad \right) \!-\!b_{22}\partial _2g_1\quad \right) e_2\nonumber \\&\quad +\left( \partial _2^{2}g_1 +\frac{b_{11}b_{22}-b_{12}^{2}}{g_1}\right) e_3. \end{aligned}$$
(103)

Now, from the existence of \(Y(x_1,x_2)\) and the Schwartz theorem \(\partial _1\partial _2Y=\partial _2\partial _1Y\), we obtain the following integrability condition:

$$\begin{aligned} \partial _2A_1-\partial _1A_2=A_1A_2-A_2A_1. \end{aligned}$$
(104)

But we have \(\partial _1A_2=0\), and

$$\begin{aligned} \partial _2A_1= \begin{pmatrix} 0&{}\partial _2^{2}g_1&{}-\partial _2(b_{11}/g_1)\\ -\partial _2^{2}g_1&{}0&{}-\partial _2(b_{12})\\ \partial _2(b_{11}/g_1)&{}\partial _2(b_{12})&{}0 \end{pmatrix} \end{aligned}$$
(105)

On another side, by computing the commutator \([A_1,A_2]=A_1A_2-A_2A_1\), we find:

$$\begin{aligned}&[A_1,A_2]\nonumber \\&\,=\begin{pmatrix} 0&{}(b_{12}^{2}-b_{11}b_{22})/g_1&{}-b_{22}\partial _2g_1\\ (b_{11}b_{22}-b_{12}^{2})/g_1&{}0&{}b_{12}\partial _2g_1/g_1 \\ b_{22}\partial _2g_1&{}-b_{12}\partial _2g_1/g_1&{}0 \end{pmatrix} \end{aligned}$$
(106)

This implies the three following conditions:

$$\begin{aligned}&\partial _2(b_{11}/g_1)=b_{22}\partial _2g_1,\quad \partial _2(b_{12}) =-b_{12}\frac{\partial _2g_1}{g_1},\nonumber \\&b_{11}b_{22}-b_{12}^{2}=-g_1\partial _2^{2}g_1. \end{aligned}$$
(107)

Together they tell precisely that \(\partial _2v=0\).

The last condition is the Theorema Egregium of Gauss, which computes the curvature \(K\) from the metric. The two other conditions are the equations of Gauss–Codazzi. The three equations together are the necessary and sufficient conditions on \(g_{ij},b_{ij}\) to be the tensors of a surface in \(\mathbb {R}^{3}\), in the case where there is a continuous symmetry \(\partial _1\).

Remark we could have established the invariance of \(v\) directly by using the integrability condition (Eq. 104), by noting that the axis of the rotation generated by \(A_1\) in a fixed frame is the kernel of \(YA_1Y^{-1}\), and observing that

$$\begin{aligned} \partial _2(YA_1Y^{-1})&= YA_2A_1Y^{-1}+Y\partial _2A_1Y^{-1}\nonumber \\&\quad -YA_1Y^{-1}(YA_2)Y^{-1}=0. \end{aligned}$$
(108)

To finish, we have to control \(r(x_1,x_2)\) from what we have learned on \(r_1\) and \(r_2\). We choose the coordinates \(x,y,z\) of the space in such a manner that \(v\) is parallel to \(0y\), and we denote by \(\theta \) the angular coordinate in the plane \(Oxz\). As the angular velocity is constant we can pose \(x_1=\theta \). In these coordinates, the vector field \(r_1\) is given by

$$\begin{aligned} r_1(x_1,x_2)&=(-X(x_2)\sin (x_1)\nonumber \\&\quad -Z(x_2)\cos (x_1), A(x_2),X(x_2)\cos (x_1)\nonumber \\&\quad -Z(x_2)\sin (x_1)); \end{aligned}$$
(109)

for smooth functions \(R,A\) of \(x_2\). Consequently, there are smooth functions \(F,G,B\) of \(x_2\) such that

$$\begin{aligned} r(x_1,x_2)&=(X(x_2)\cos (x_1)-Z(x_2)\sin (x_1)\nonumber \\&\quad +F(x_2), A(x_2)x_1+B(x_2), X(x_2)\sin (x_1)\nonumber \\&\quad +Z(x_2)\cos (x_1)+G(x_2)). \end{aligned}$$
(110)

This gives

$$\begin{aligned} r_2(x_1,x_2)&= (X'(x_2)\cos (x_1)-Z'(x_2)\sin (x_1)\nonumber \\&\quad +F'(x_2),A'(x_2)x_1+B'(x_2),X'(x_2)\sin (x_1)\nonumber \\&\quad +Z'(x_2)\cos (x_1)+G'(x_2)), \end{aligned}$$
(111)

where a prime denotes the derivative with respect to \(x_2\). Then

$$\begin{aligned} g_{12}&=r_1.r_2=Z'X-ZX'+(-XF'-ZG')\sin (x_1)\nonumber \\&\quad +(XG'-ZF')\cos (x_1)+AA'x_1+AB'. \end{aligned}$$
(112)

Thus, \(g_{12}=0\) implies \(F'=G'=A'=0\), \(AB'=ZX'-XZ'\). Then, we can translate the coordinates in \(Oxz\) in such a manner that \(F=G=0\), which we assume now. And \(A\) is a constant.

We note the additional constraint:

$$\begin{aligned} AB'=ZX'-XZ', \end{aligned}$$
(113)

which tells that \(AB(x_2)\) measures the area enclosed by the arc of the plane curve \((X(x_2),Z(x_2))\).

Now

$$\begin{aligned} g_{22}=r_2 \cdot r_2=X'^{2}+Z'^{2}+B'^{2}. \end{aligned}$$
(114)

Thus, to normalize by \(g_{22}=1\), we have to choose for \(x_2\) the curvilinear abscissa of the space curve \(\varGamma \) of coordinates \((X,B,Z)\).

If \(A\) and \(B\) are nonzero, we obtain for \(\varSigma \) the surface generated by the motion of \(\varGamma \) under the twists directed by \(Oy\) with angle \(x_1=\theta \) and translation \(A\theta \). The constraint (Eq. 113) insures that the generating helices are perpendicular to \(\varGamma \) at their intersection points. Then \(\varSigma \) is a generalized helicoid.

In the case where \(A=0\), but \(B\ne 0\), \(\varSigma \) is a surface of revolution, and it exists a function \(R(y)\) such that we can write:

$$\begin{aligned} r(\theta ,y)=(R(y)\cos \theta ,y,R(y)\sin \theta ). \end{aligned}$$
(115)

In the case where \(A\) is a nonzero constant, but where \(B\) is a constant, we can annihilate \(B\) by a translation along \(Oy\). In this case \((X,Z)\) satisfies \(XZ'=ZX'\), thus it is a straight line through \(0\); then we can take the distance \(R\) to \(0\) as the coordinate \(x_2\), and \(\varSigma \) is the standard helicoid.

$$\begin{aligned} r(\theta ,R)=(-R\sin \theta ,A\theta ,R\cos \theta ). \end{aligned}$$
(116)

\(\square \)

Proof of Proposition B

The demonstration of (i) is easy: first, for any cylinder, the normal \(n\) is invariant by \(\partial _1\), thus \(b_{11}=0\), and the Gauss curvature is constant, thus \(b_{12}=0\). We could also have used that \(\partial _2g_1=0\) and the Eq. (107). So, if the cylinder is minimal, we have \(H=0\) and then \(b_{22}=0\), which gives the plane.

For (iii), we recall the direct proof that the surfaces of revolution in \(\mathbb {R}^{3}\), which are minimal surfaces are the catenoids:

Any surface of revolution is given by the following equations:

$$\begin{aligned}&x(\theta ,y)=R(y)\cos \theta , \nonumber \\&y(\theta ,y)=y,\nonumber \\&z(\theta ,y)=R(y)\sin \theta . \end{aligned}$$
(117)

Easy computations verify that

$$\begin{aligned} r_\theta \cdot r_\theta =R^{2},\quad r_x \cdot r_y=0,\quad r_y \cdot r_y=1+R_y^{2}. \end{aligned}$$
(118)

The unit normal vector is

$$\begin{aligned} n=-\frac{1}{\sqrt{1+R_y^{2}}}(\cos \theta ,R_y,\sin \theta ). \end{aligned}$$
(119)

Thus we have

$$\begin{aligned} b_{\theta \theta }=R/\sqrt{1+R_y^{2}},\quad b_{\theta y}=0,\quad b_{yy}=-R_{yy}/\sqrt{1+R_y^{2}}.\nonumber \\ \end{aligned}$$
(120)

Which gives

$$\begin{aligned} b_{\theta }^{\theta }=1/R\sqrt{1+R_y^{2}},\quad b_{y}^{y}=-R_{yy}\sqrt{1+R_y^{2}}. \end{aligned}$$
(121)

To have a minimal surface, we must have \(b_{\theta }^{\theta }+b_{y}^{y}=0\), that is the differential equation

$$\begin{aligned} RR_{yy}=1+R_y^{2}. \end{aligned}$$
(122)

All solutions are given by the formula

$$\begin{aligned} R(y)=a\cosh \left( \frac{y-y_0}{a}\right) , \end{aligned}$$
(123)

where \(a,y_0\) are any positive real constants. \(\square \)

The proof of the general case (ii) is more elaborated. We begin by using the Eq. (107) to compute directly \(g_1(x_2)\).

The first equation \(\partial _2(b_{11}/g_1)=b_{22}\partial _2g_1\), can be written

$$\begin{aligned} (b_1^{1}-b_2^{2})\partial _2g_1+g_1\partial _2b_1^{1}=0. \end{aligned}$$
(124)

By adding the hypothesis of minimality \(b_1^{1}+b_2^{2}=0\), this gives

$$\begin{aligned} b_1^{1}\frac{1}{g_1}\partial _2g_1=-2\partial _2b_1^{1}. \end{aligned}$$
(125)

Thus it exists a constant \(C_1\) such that \(b_1^{1}=C_1/g_{11}\), which can also be expressed by

$$\begin{aligned} b_{11}=C_1. \end{aligned}$$
(126)

Which gives \(b_{22}=-C_1/g_{11}\). The second equation \(\partial _2(b_{12})=-b_{12}\frac{\partial _2g_1}{g_1}\) gives the existence of a second constant \(C_2\) such that

$$\begin{aligned} b_{12}=C_2/g_1. \end{aligned}$$
(127)

Now the third equation \(b_{11}b_{22}-b_{12}^{2}=-g_1\partial _2^{2}g_1\) is equivalent to the second order differential equation

$$\begin{aligned} \partial _2g_1=(C_1^{2}+C_2^{2})g_1^{-3}. \end{aligned}$$
(128)

This equation is easy to integrate: all positive solutions are given by the formula

$$\begin{aligned} g_1=\sqrt{a^{2}+b^{2}(x_2-c)^{2}}, \end{aligned}$$
(129)

where \(a,b,c\) are arbitrary constants, just linked by the constraint \(C_1^{2}+C_2^{2}=a^{2}b^{2}\).

From here we compute the Gauss curvature

$$\begin{aligned} K=\frac{-a^{2}b^{2}}{(a^{2}+b^{2}(x_2-c)^{2})^{2}} \end{aligned}$$
(130)

Note that \(g_2=1\).

In what follows, we assume \(c=0\), which corresponds to an inoffensive translation in \(x_2\). And we come back to the Eq. (110) for the surface \(\varSigma \), taking in account the conditions \(F=G=A'=0\).

If \(B'=0\), a translation along \(Oy\) permits to assume that \(B=0\); then the curve \(\varGamma \) belongs to a line through \(0\). The surface is a ruled helicoid, parameterized by \(x_2=R,x_1=\theta \); as we already saw at the end of the proof of the Proposition 7: \(x=-R\sin \theta , y=A\theta , z=R\cos \theta \). We have \(r_1=(-R\cos \theta ,A,-R\sin \theta )\), and \(r_2=(-\sin \theta ,0,\cos \theta )\), thus \(g_{11}=R^{2}+A^{2}\), \(g_{12}=0\), \(g_{22}=1\). Which gives \(n=(A\cos \theta ,R,A\sin \theta )/g_1\), thus \(b_{11}=b_{22}=0\), and \(b_{12}=-A/g_1\). We thus verified that this ruled helicoid is a minimal surface. The parameters satisfy \(a^{2}=A^{2},b^{2}=1\).

Now we suppose that \(B'\ne 0\). (We assume this condition holds for one value of \(x_2\), but we will see that it holds true for any value of \(x_2\)). We introduce a new coordinate \(t\) in place of \(x_2\), such that \(B(x_2)=B_0t\). The coordinates \((x_1,t)\) are orthogonal, but not normalized. In what follows the prime will denote the derivative with respect to \(t\).

We have

$$\begin{aligned}&r_1=(-X\sin x_1-Z\cos x_1,A,X\cos x_1-Z\sin x_1)\nonumber \\&r_2=\frac{(X'\cos x_1-Z'\sin x_1,B_0,X'\sin x_1+Z'\cos x_1)}{x'_2} \end{aligned}$$
(131)

Note that \(t'=1\). The orthogonality of coordinates implies \(ZX'-XZ'=AB_0\). We have \(g_{11}=X^{2}+Z^{2}+A^{2}\), which gives:

$$\begin{aligned} a^{2}+b^{2}x^{2}_2=X^{2}+Z^{2}+A^{2} \end{aligned}$$
(132)

thus by differentiation

$$\begin{aligned} b^{2}x_2x'_2=XX'+ZZ' \end{aligned}$$
(133)

The unit normal vector \(n\) is the normalized cross product \((r_1\times r_2)/\Vert r_1\times r_2\Vert \), thus it is also equal to \(N/\Vert N\Vert \), where

$$\begin{aligned} N&= ((AZ'-B_0X)\cos x_1+(AX'+ZB_0)\nonumber \\&\times \sin x_1,XX'+ZZ',\nonumber \\&-(AZ'-B_0X)\sin x_1-(AX'+ZB_0)\cos x_1) \end{aligned}$$
(134)

We have

$$\begin{aligned}&r_{11}=(-X\cos x_1+Z\sin x_1,0,-X\sin x_1-Z\cos x_1),\nonumber \\&r_{12}=\frac{(-X'\sin x_1-Z'\cos x_1,0,X'\cos x_1-Z'\sin x_1)}{x'_2}. \end{aligned}$$
(135)

Thus

$$\begin{aligned} b_{11}&=r_{11} \cdot n\!=\!(-X(AZ'-B_0X)\!+\!Z(AX'\!+\!ZB_0))/\Vert N\Vert \nonumber \\&=(B_0(X^{2}+Z^{2})+A(ZX'-XZ'))/\Vert N\Vert \nonumber \\&=(B_0(X^{2}+Z^{2}+A^{2}))/\Vert N\Vert \nonumber \\&=B_0g_{11}/\Vert N\Vert \end{aligned}$$
(136)

and

$$\begin{aligned} b_{12}&=r_{12} \cdot n\!=\!(-Z'(AZ'-B_0X)\nonumber \\&\quad -X'(AX'+ZB_0))/{(}x'_2\Vert N\Vert {)} \nonumber \\&=(-B_0(ZX'-XZ')-A(X'^{2}+Z'^{2}))/{(}x'_2\Vert N\Vert {)} \nonumber \\&=-A(X'^{2}+Z'^{2}+B_0^{2})/{(}x'_2\Vert N\Vert {)} \nonumber \\&=-Ax'_2/\Vert N\Vert . \end{aligned}$$
(137)

We saw that for a minimal helicoid it exist constants \(C_1,C_2\) such that \(b_{11}=C_1\) and \(b_{12}=C_2g_1\); thus

$$\begin{aligned} x'_2=-\frac{C_2B_0}{C_1A}g_1,\quad \Vert N\Vert =\frac{B_0}{C_1}g_1^{2}. \end{aligned}$$
(138)

In addition we can compute

$$\begin{aligned} \Vert N\Vert ^{2}&=(AZ'\!-\!B_0X)^{2}\!+\!(AX'\!+\!ZB_0)^{2}+(XX'+ZZ')^{2}\nonumber \\&=A^{2}(X'^{2}+Z'^{2})+B_0^{2}(X^{2}+Z^{2})+(XX'+ZZ')^{2}\nonumber \\&\quad +2AB_0(ZX'-XZ')\nonumber \\&=A^{2}((x'_2)^{2}-B_0^{2})+B_0^{2}(g_{11}-A^{2}) +b^{4}(x_2)^{2}(x'_2)^{2}\nonumber \\&\quad +2A^{2}B_0^{2}\nonumber \\&=x_2^{4}b^{6}\frac{C_2^{2}B_0^{2}}{C_1^{2}A^{2}} +x_2^{2}\left( \frac{C_2^{2}B_0^{2}}{C_1^{2}A^{2}}(b^{4}a^{2}+b^{2}A^{2}) +B_0^{2}b^{2}\right) \nonumber \\&\quad +a^{2}B_0^{2}\left( 1+\frac{C_2^{2}}{C_1^{2}}\right) . \end{aligned}$$
(139)

By comparing with

$$\begin{aligned} \Vert N\Vert ^{2}=B_0^{2}C_1^{-2}(b^{4}x_2^{4}+2a^{2}b^{2}x_2^{2}+a^{4}), \end{aligned}$$
(140)

we find

$$\begin{aligned}&b^{2}=\frac{A^{2}}{C_2^{2}},\nonumber \\&a^{2}=C_2^{2}+C_1^{2},\nonumber \\&C_2^{2}(b^{2}a^{2}+A^{2})+C_1^{2}A^{2}=2a^{2}A^{2}; \end{aligned}$$
(141)

but we already known that \(b^{2}a^{2}=C_2^{2}+C_1^{2}\), and we can assume \(a,b\) positive, so this implies

$$\begin{aligned} b=1,\quad A^{2}=C_2^{2},\quad a=\sqrt{C_2^{2}+C_1^{2}}. \end{aligned}$$
(142)

Moreover the formula for \(x'_2\) gives

$$\begin{aligned} -Cdt=\frac{\mathrm{d}x_2}{\sqrt{a^{2}+x_2^{2}}}, \end{aligned}$$
(143)

for \(C=B_0C_2/AC_1\); thus, assuming \(x_2(0)=0\), we get

$$\begin{aligned} x_2=-a\sinh (Ct), \end{aligned}$$
(144)

This implies

$$\begin{aligned} x'_2=-Ca\cosh (Ct),\quad x_2x'_2=a^{2}C\cosh (Ct)\sinh (Ct)\nonumber \\ \end{aligned}$$
(145)

Which gives the differential system for \(X(t),Z(t)\):

$$\begin{aligned} ZX'-XZ'&=AB_0\nonumber \\ XX'+ZZ'&=a^{2}C\cosh (Ct)\sinh (Ct). \end{aligned}$$
(146)

Note this system is regular in the polar coordinates \((\varTheta ,R)\), where \(\tan \varTheta =Z/X,R^{2}=X^{2}+Z^{2}\), when \(R\ne 0\) is insured. It can be written:

$$\begin{aligned} \varTheta '&=-R^{2}AB_0\nonumber \\ R'&=\frac{a^{2}C}{R}\cosh (Ct)\sinh (Ct), \end{aligned}$$
(147)

which permits an easy integration, by computing first \(R(t)\) then \(\varTheta (t)\) from it. However, thanks to Scherk formulas (1834), we have a direct candidate for the solution of the system (Eq. 146), when the initial condition are \(X(0)=0, Z(0)\ne 0\), which we can assume after rotation in the plane \(x,z\), corresponding to an inoffensive translation in \(x_1\); this is:

$$\begin{aligned} X(t)&=\alpha \cosh Ct,\nonumber \\ Z(t)&=\gamma \sinh Ct. \end{aligned}$$
(148)

Due to the formula \(\cosh ^{2} Ct-\sinh ^{2} Ct=1\), the first equation of the system is satisfied if and only if \(\alpha \gamma C=-AB_0\), i.e., \(\alpha \gamma B_0C_2/AC_1=-AB_0\), i.e., \(\alpha \gamma =-C_1C_2\). And the second equations is satisfied if and only if \(\alpha ^{2}+\gamma ^{2}=a^{2}\), i.e., \(\alpha ^{2}+\gamma ^{2}=C_1^{2}+C_2^{2}\).

Observe that from the beginning, \(B_0\) is arbitrary (if different from \(0\)), corresponding to a linear change of the variable \(t\), so we can choose \(B_0=-AC_1/C_2\) (note \(C_1,C_2\) are intrinsic quantities), thus \(C=-1\), and we can put \(A=C_2\),\(\alpha =C_1\),\(\gamma =-C_2\). This gives the beautiful surface:

$$\begin{aligned} x&=C_1\cosh t\cos x_1-C_2\sinh t\sin x_1\nonumber \\ y&=C_2x_1-C_1t\nonumber \\ z&=C_1\cosh t\sin x_1+C_2\sinh t\cos x_1. \end{aligned}$$
(149)

We have established that for any constants \(C_1,C_2\), this surface has a first and second tensor invariant by \(\partial _1\), that its normalized coordinate \(x_2\) satisfies \(g_{11}=C_1^{2}+C_2^{2}+x_2^{2}\) and that we have \(b_{11}=C_1,b_{12}=C_2/g_1\). Thus, it is minimal by the Gauss–Codazzi equations. Moreover, we have establish that up to isometry of \(\mathbb {R}^{3}\), this surface is the only invariant minimal surfaces with \(b_{11}=C_1,b_{12}=C_2/g_1\), because the system (Eq. 146) has a unique solution with given initial value of \((X(0),Z(0))\) different from \((0,0)\).

Remark the slopes \(\pm C_1\) of \(y\), with respect to \(t\) in the twisted curve \(\varGamma \), are the only ones giving an arc-length \(x_2\), which is rational in \(\sinh t,\cosh t\); the other values would have given elliptic integrals. This offers an explicit formula in \(x_2\) for \(\varSigma \):

$$\begin{aligned} x&=C_1\sqrt{1+\frac{x_2^{2}}{C_1^{2}+C_2^{2}}}\cos x_1+\frac{C_2x_2}{\sqrt{C_1^{2}+C_2^{2}}}\sin x_1\nonumber \\ y&=C_2x_1+C_1 \ln \left( 1+2\frac{x_2}{\sqrt{C_1^{2}+C_2^{2}}}\right) \nonumber \\ z&=C_1\sqrt{1+\frac{x_2^{2}}{C_1^{2}+C_2^{2}}}\sin x_1-\frac{C_2x_2}{\sqrt{C_1^{2}+C_2^{2}}}\cos x_1. \end{aligned}$$
(150)

\(\square \)

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Marianelli, P., Berthoz, A. & Bennequin, D. Crista egregia: a geometrical model of the crista ampullaris, a sensory surface that detects head rotations. Biol Cybern 109, 5–32 (2015). https://doi.org/10.1007/s00422-014-0623-5

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