## Abstract

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

## 1. Introduction

In a zero-gravity environment, the free surface 𝒮 of a liquid droplet in equilibrium has a constant mean curvature. This physical property motivates a classical problem in differential geometry: the search for all surfaces with constant mean curvature. One such surface could be either *embedded* or *immersed*, depending on the regularity of the immersion of the surface in ^{3}; in particular, an immersed surface may possess self-intersections, whereas an embedded surface cannot. For a very long time, the only known examples of constant mean curvature surfaces were spheres, cylinders and the family of axis-symmetric surfaces discovered by Delaunay (1841). The first constant mean curvature surfaces different from these were discovered by Wente (1986), while Kapouleas (1990) constructed a plethora of such surfaces employing pieces of spheres and pieces of Delaunay's surfaces.

For free droplets in space, the sphere is clearly a possible equilibrium shape. Actually, Alexandrov (1958, 1962*a*,*b*) proved that the sphere is the only closed, embedded surface with constant mean curvature and hence a free droplet in equilibrium must be a ball. The theorem of Alexandrov is a partial answer to the more general problem formulated by lifting the embedding regularity hypothesis; this problem had received a tentative solution by *Hopf's conjecture*. In 1946, Hopf conjectured that a constant mean curvature characterizes the sphere among all closed immersed surfaces, having proved that this is the case in the subset of surfaces with genus ` g`=0 (see Hopf 1983). It should be noted that in Hopf's conjecture, no hypothesis was made either on the regularity or on the genus of the surface, while these hypotheses are crucial to characterize the sphere in both Alexandrov's and Hopf's theorems. Forty years later, Wente (1986) disproved Hopf's conjecture, providing examples of closed, immersed surfaces with constant mean curvature of genus

`=1, now known as`

*g**Wente's tori*. Other closed, immersed surfaces with genus

`≥2 were constructed by Kapouleas (1992) by fusing together different Wente's tori.`

*g*In this paper, we prove that a closed, immersed surface with constant mean curvature must satisfy three integral criteria that follow from requiring the second variation of the area functional to be frame indifferent, much in the spirit of the integral identity derived in a previous study on the equilibrium of lipid vesicles in two space dimensions (see Rosso *et al.* 2003, §3). These criteria are derived in §2. In §3, we apply them to characterize the sphere as the only closed axis-symmetric surface with constant mean curvature (Delaunay 1841). Our new proof of this old result adopts as surface parametrizations the solutions to a certain partial differential equation. Though this equation is heavily exploited here, its solutions are not explicitly involved in the proposed criteria, at variance with the studies of Wente (1986) and Bobenko (1991), where these solutions are indeed needed to prove that the resulting surfaces may be closed. The paper ends with appendix A, where the elementary steps of a lengthy computation required in §3 are recorded.

## 2. Criteria

The criteria we propose in this paper stem from a simple remark on the second variation of the area functional. This functional has a classical physical motivation which has actually inspired our remark.

Consider a liquid droplet with assigned volume *V* in the absence of gravity. Let ℬ be the region in space occupied by the droplet and let 𝒮 denote its boundary, assumed to be an orientable, sufficiently smooth surface. The energy functional ℱ that describes the droplet iswhere *γ*>0 is the surface tension of the liquid in its environment and *a* denotes the area measure. ℱ is subject to the constraint(2.1)where *v* is the volume measure. Finding stable equilibrium configurations for the droplet amounts to minimizing ℱ among all admissible sets ℬ that satisfy (2.1). The solution to this problem requires computing both the first and the second variation of ℱ. In the presence of a constraint like (2.1), the perturbations of ℬ that induce both *δ*ℱ and *δ*^{2}ℱ must be constrained so as to maintain the constraint up to second order in the perturbation parameter. As remarked by Barboso & do Carmo (1984) in their stability analysis for (hyper)surfaces with constant mean curvature (CMC), ‘if perturbations were not properly selected, spheres would not be stable’. Thus, as done by Rosso & Virga (2004), we introduce the fields * u* and

*on 𝒮 so that every point*

**v***p*of 𝒮 is transformed into(2.2)where

*ϵ*is a perturbation parameter. The deformed surface 𝒮

_{ϵ}, which bounds the deformed body ℬ

_{ϵ}, is the collection of all

*p*

_{ϵ}as in (2.2) with

*p*spanning 𝒮. Both fields

*and*

**u***are appropriately constrained for ℬ*

**v**_{ϵ}to satisfy (2.1) up to second order in

*ϵ*.

The first variation of ℱ is a linear functional of the normal component *u* of * u*, which reads as(2.3)where

*H*is the

*total*curvature of 𝒮, i.e. twice its mean curvature, and

*λ*is the Lagrange multiplier associated with the constraint (2.1). Formally,where

*is the unit normal to 𝒮 oriented outward from ℬ and div*

**ν**_{s}denotes the surface divergence. It readily follows from (2.3) that the stationary surfaces 𝒮 have CMC: they represent equilibrium shapes of weightless liquid droplets subject to surface tension. The stability analysis of these equilibria builds upon the form taken on them by the second variation

*δ*

^{2}ℱ,(2.4)where ∇

_{s}denotes the surface gradient and

*K*is the Gaussian curvature.

*δ*

^{2}ℱ is a quadratic functional subject to(2.5)which enforces the requirement (2.1) on the volume of the droplet. While it is clear that

*δ*ℱ could only depend on

*u*, it is remarkable that

*δ*

^{2}ℱ also depends on

*u*only, and not on

*, once evaluated upon the surfaces 𝒮 that make*

**v***δ*ℱ=0.

Upon perturbing a closed CMC surface 𝒮 without affecting the volume it encloses, the functional ℱ changes as follows:(2.6)Our major argument here is frame indifference. If the perturbation that changes 𝒮 into 𝒮_{ϵ} is an infinitesimal rigid displacement **u**_{R}, then(2.7)since ℱ is invariant under change of frame. By combining (2.7) and (2.6), one readily arrives at(2.8)where *u*_{R}:=**u**_{R}.* ν*. The interesting fact is that while

*δ*ℱ[

*u*

_{R}] vanishes identically for every surface 𝒮, be it with CMC or not (see proposition 2.1), equation (2.8) is by no means an identity when

*δ*

^{2}ℱ is expressed in the form (2.4), as this is only valid under the assumption that 𝒮 be a closed CMC surface. Thus, requiringto hold for all rigid displacements

**u**_{R}will entail necessary integral criteria valid for all closed CMC surfaces.

To capture by this strategy the features that all CMC surfaces must possess, we can no longer ignore a central technical issue: the surface regularity. We suppose that 𝒮 can locally be represented by a *C*^{∞}-mapping *F* of a domain *U* in ^{2} onto ^{3}. We further assume that the differential ∇*F* is injective at every point of *U* and realizes a homeomorphism onto its image. This, together with the global requirement that 𝒮 be closed, amounts to say that 𝒮 is a closed, compact, orientable surface (Lemaire 2003).

The local properties of 𝒮 may or may not be accompanied by the global injectivity of the map resulting from patching together all different local mappings *F*'s: in the case that it is, 𝒮 has no self-intersections, whereas in the case that it is not, 𝒮 may have self-intersections. Often, in the former case, 𝒮 is said to be *embedded* in ^{3}, whereas in the latter case, it is said to be *immersed*. In the more precise language of Hopf (1983), 𝒮 would correspondingly be a *simple* closed surface and a *general* closed surface, respectively.

A simple closed surface is orientable and possesses a unique interior; thus, strictly speaking, it would be the most natural definition for the boundary of a droplet. A theorem of Alexandrov (1958, 1962*a*,*b*) shows that among all simple closed surfaces only the sphere has CMC. The search for general closed surfaces with CMC has been far more exciting (Lemaire 2003). Wente (1986) proved that there exists a CMC torus immersed in ^{3}. Kapouleas (1987, 1990, 1991, 1992, 1995) proved that there exist closed CMC surfaces immersed in ^{3} of any genus ` g`≥2.

Though general closed surfaces have no uniquely defined interior, the notion of enclosed volume can also be extended to them (Hopf 1983, p. 135). Both equations (2.3) and (2.4) can accordingly be extended to this class of surfaces. Thus, the request of invariance for *δ*^{2}ℱ under rigid displacements still applies, though the physical interpretation of 𝒮 as boundary of a droplet may well be lost.

Before stating our main result, we show for completeness that *δ*ℱ[*u*_{R}] would also vanish for a surface 𝒮 that fails to be stationary for ℱ.

*The first variation δ*ℱ *of* ℱ *vanishes identically for*(2.9)*where* _{0} *is a translation and* *is the skew tensor that generates a rotation about the origin o*.

It follows from (2.9) thatBy the surface divergence theorem (e.g. Virga 1984, section 2.3),(2.10)Similarly, by the divergence theorem,(2.11)Making use of both (2.10) and (2.11) in (2.3), we prove the assertion. ▪

*A similar conclusion does not apply to δ*^{2}ℱ *as given in* *(2.4)**,* *since this formula applies only to closed CMC surfaces. As shown in appendix C of* *Rosso* & *Virga (2004)**,* *if* 𝒮 *is not closed*, *δ*^{2}ℱ *acquires an integral along the border of* 𝒮, *while if* 𝒮 *has no CMC, the integral over* 𝒮 *in* *(2.4)* *acquires, among others, a contribution depending on* ∇_{s}*H*.

*It should also be noted that, by* *(2.11)*, **u**_{R} *satisfies* *(2.5)**,* *and so it is an admissible field for δ*^{2}ℱ *in* *(2.4)*.

Requiring *δ*^{2}ℱ to vanish for *u*_{R}=**u**_{R}.* ν*, we now prove theorem 2.4.

If 𝒮 *is a general closed CMC surface, the following conditions hold*:(2.12)(2.13)(2.14)*where o is the origin in space*; *is the identity tensor; and* *and* *are the skew tensors associated with ν and* ≔(

*p*−

*o*),

*respectively*.

For any pair of vectors * a* and

*,*

**b***.*

**a***=*

**b***.*

**w***×*

**a***, where*

**b***is the vector associated with the skew tensor . Thus,*

**w***u*

_{R}is linear in both

**u**_{0}and

*. It then follows from (2.4) that*

**w***δ*

^{2}ℱ[

*u*

_{R}] is a quadratic form in the pair (

**u**_{0},

*), which can be given the general representation(2.15)where*

**w**_{1}(𝒮),

_{2}(𝒮) and

_{3}(𝒮) are tensors depending only on 𝒮. For

*δ*

^{2}ℱ[

*u*

_{R}] to vanish identically for all (

**u**_{0},

*), all these tensors must be zero.*

**w**To identify _{1}(𝒮), we set * w*=

**0**in (2.15). A simple computation then leads us to(2.16)It follows from (2.16) that the tensor

_{1}(𝒮) is defined as in (2.12).

Likewise, we set **u**_{0}=**0** and evaluate *δ*^{2}ℱ[*u*_{R}]. Recalling thatwe arrive at(2.17)where * r*=(

*p*−

*o*). To convert (2.17) into a quadratic form in

*, we recall a property of skew tensors,(2.18)where is a generic tensor, and and are the skew tensors associated with and , respectively. Repeated use of (2.18) in (2.17) allows us to identify up to a sign the tensor*

**w**_{2}(𝒮) in (2.13).

Finally, when neither **u**_{0} nor * w* vanishes, subtracting from (2.15) the quadratic forms associated with

_{1}(𝒮) and

_{2}(𝒮), we readily extract the bilinear form(2.19)Recalling that , by a direct computation we write the tensor associated with (2.19) as

_{3}(𝒮) in (2.14). This concludes the proof of theorem 2.4. ▪

*The tensor* *in both* *(2.13)* *and* *(2.14)* *suggests that both* _{2}(𝒮) *and* _{3}(𝒮) *also depend on the origin o*. *This is indeed the case, but the requirement that* _{1}(𝒮), _{2}(𝒮) *and* _{3}(𝒮) *vanish is independent of the choice of the origin. In other words, once equations* *(2.12)*–*(2.14)* *are valid for a given point o, they are so for all such points*.

To show this, we move *o*′ into *o* by the translation * a*=

*o*−

*o*′, so thatCorrespondingly,(2.20)Denoting by , and the tensors defined in (2.12)–(2.14), but with

*o*replaced by

*o*′ in (2.13) and (2.14), by (2.20), we arrive atIt is clear from these equations that , and vanish whenever

_{1}(𝒮),

_{2}(𝒮) and

_{3}(𝒮) do, whatever may be . ▪

## 3. Application

The criteria derived in §2 are necessary conditions for the existence of *closed* CMC surfaces, and so they can typically be applied to rule out the existence of such surfaces within a certain class. Here, as an illustration of our method, we apply these criteria to characterize the sphere as the only closed CMC surface among all axis-symmetric surfaces. This result is indeed a special case of a theorem by Delaunay (1841). We give here a new proof of this old result and, more importantly, assess the effectiveness of the proposed criteria in discerning closed CMC surfaces from non-compact ones.

Before proceeding further, we recall a few preliminary, technical details to be employed in our proof below. A parametrization * r*(

*u*,

*v*) of the surface 𝒮 is

*conformal*whenever(3.1)where

**r**_{u}≔∂

_{u}

*and*

**r**

**r**_{v}≔∂

_{v}

*. The scalar parameters (*

**r***u*,

*v*) for 𝒮 are not to be confused with the vector fields (

*,*

**u***) appearing in equation (2.2). If*

**v***(*

**r***u*,

*v*) is a conformal parametrization, then it satisfies the following equation:(3.2)where

*Δ*

*=*

**r**

**r**_{uu}+

**r**_{vv}. Hence, if

*H*is constant, then the solutions to (3.2) are conformal parameterizations of CMC surfaces. Every surface has a conformal parametrization (e.g. Bianchi 1927). In particular, for an axis-symmetric surface, there exist two functions

*f*(

*u*) and

*g*(

*u*), the former of which is positive, such that(3.3)It is readily seen that this representation enjoys the property(3.4)We are now in a position to prove theorem 3.1.

*Let* 𝒮 *be an axis-symmetric CMC surface*. 𝒮 *is closed if and only if it is a sphere*.

Since 𝒮 is an axis-symmetric CMC surface, we can assume that *H*≠0. With no loss of generality, *H* can be rescaled to *H*=2. Moreover, for the conformal parametrization (3.3), conditions (3.1) and (3.2) become(3.5)and(3.6a)(3.6b)respectively. Both (3.6*a*) and (3.6*b*) have a first integral, and (3.5) reduces to a relation between the corresponding integration constants. A direct calculation then yields(3.7a)(3.7b)where *c*≥0 is the independent integration constant. The right-hand side of equation (3.7*a*) can be rewritten in the form(3.8)The positive roots of this polynomial, which are also the values attainable by the function *f*(*u*) at its turning points, areAll solutions to (3.7*a*) oscillate periodically between these values and are transformed into one another by a translation in *u*. Among these equivalent realizations, we choose the function *f*(*u*) even in *u*, that has its maximum at and minimum at ; its inverse is represented asThe only closed surface of genus ` g`=0 in this class is the sphere. If we require

*f*

_{m}=0, we are led to

*c*=0; in turn, the unique solution of (3.7

*a*) and (3.7

*b*) is Mercator's representation of a sphere (Wente 2001).

We now evaluate _{1}(𝒮), _{2}(𝒮) and _{3}(𝒮) for all *c*≥0. Recalling that *g*(*a*)=*g*(−*a*)=0, with repeated use of (3.6*a*, 3.6*b*) and (3.7*a*, 3.7*b*), integrations by parts and the change of variable , we arrive at(3.9)where(3.10)All steps relevant to the lengthy computation that led us from (2.13) to (3.9) are recorded in appendix A. Tensors _{1}(𝒮) and _{3}(𝒮) vanish identically as a consequence of the evenness of *f*; _{2}(𝒮) is instead a function of the integration constant *c*. We remark that, consistent with the preceding discussion, *Ψ*(0)=0. To complete our proof, we need to establish that *c*=0 is the only zero of *Ψ*(*c*); then only the sphere would satisfy all the three criteria. To obtain an explicit formula for *Ψ*(*c*), we first treat (*c*,*f*) as independent variables and change them to(3.11)We then rewrite *Ψ*(*c*) as(3.12)where *P*(*x*, *y*) is the integrand in (3.10); *K* is the complete elliptic integral of the first kind; and _{2}*F*_{1} is a hypergeometric function (e.g. Gradshteyn & Ryzhik 1994). We can easily recover from (3.12) the asymptotic expansion of *Ψ*(*c*) as *c* →∞: since the third term of the series in this asymptotic expansion is negative, employing only the first two terms we define the functionwhich is greater than *Ψ*(*c*) for sufficiently large *c*. Moreover, *Ψ*(*c*)<Ψ_{∞}(*c*)<0 for all *c*>3/32. Finally, since it is possible to check numerically that *Ψ*(*c*)≤−*c*^{2} if 0≤*c*≤3/32, we conclude that *Ψ*(*c*)=0 if and only if *c*=0. Hence, the only closed, axis-symmetric CMC surface is the sphere. ▪

*In the preceding proof, equation* *(3.2)* *and its first integrals* *(3.7a, 3.7b)* *are employed to ease the computation of the three criteria, but explicit solutions to* *(3.2)* *are by no means needed*.

Bobenko (1991) solves explicitly the partial differential equation involved in constructing Wente's tori and hence introduces explicit representations for CMC surfaces, which reduce to Wente's tori whenever they are compact. Moreover, he introduces a necessary and sufficient condition for these surfaces to be closed CMC surfaces. This condition is based on an argument of algebraic topology: it would be interesting to compare the criteria proposed here with the condition introduced by Bobenko (1991).

## Footnotes

- Received October 2, 2006.
- Accepted January 8, 2007.

- © 2007 The Royal Society