THE PERIODIC ISOPERIMETRIC PROBLEM · 2010. 10. 8. · THE PERIODIC ISOPERIMETRIC PROBLEM LAURENT HAUSWIRTH, JOAQU´IN PEREZ, PASCAL ROMON & ANTONIO ROS´ Abstract. Given a discrete

THE PERIODIC ISOPERIMETRIC PROBLEM

LAURENT HAUSWIRTH, JOAQUIN PEREZ, PASCAL ROMON & ANTONIO ROS

Abstract. Given a discrete group G of isometries of R3, we study the G-isoperimetric problem,which consists of minimizing area (modulo G) among surfaces in R3 which enclose a G-invariantregion with prescribed volume fraction. If G is a line group, we prove that solutions are eitherfamilies of round spheres or right cylinders. In the doubly periodic case we prove that for mostrank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank twolattices we show, among other results, an isoperimetric inequality in terms of the topology of theisoperimetric surfaces. Finally, we study the case where G = Pm3m (the group of symmetriesof the integer rank three lattice Z3) and other crystallographic groups of cubic type. We provethat isoperimetric solutions must be spheres if the prescribed volume fraction is less than 1/6,and we give an isoperimetric inequality for G-invariant regions that, for instance, implies thatthe area (modulo Z3) of a surface dividing the three space in two G-invariant regions with equalvolume fractions, is at least 2.19 (the conjectured solution is the classical P Schwarz triply periodicminimal surface whose area is ! 2.34). Another consequence of this isoperimetric inequality is thatPm3m-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the latticegroup Z3.

Mathematics Subject Classification: 53A10, 53C42.

1. Introduction

The periodic isoperimetric problem is one of the nicest open questions in classicaldi!erential geometry: given a discrete group G of isometries of the Euclidean threespace, it consists of describing, among G-invariant regions " in R3 with prescribedvolume (modulo G), those whose boundary has least area (modulo G). The periodicisoperimetric problem is also the simplest mathematical model to explain certainshapes appearing in a number of nanostructured interface phenomena in materialsscience, where spherical, cylindrical and lamellar configurations alternate with moresophisticated bicontinuous ones. In these last cases, interfaces are small perturba-tions of triply periodic constant mean curvature Gyroids (G = I4132), Primitive(G = Pm3m) or Diamond (G = Fd3m) Schwarz surfaces and other conjecturalcandidates to solving the G-periodic isoperimetric problem for various crystallo-graphic groups G, see for instance [8, 37, 39] and, in particular, Hyde [20]. Despiteits great interest, both in geometry and crystalline surface science, the problemremains mostly unsolved.

L. Hauswirth and P. Romon are partially supported by Picasso program 02669WB and J. Perezand A. Ros by MCYT-FEDER research projects BFM2001-3318 and HF2000-0088.

1

2 LAURENT HAUSWIRTH, JOAQUIN PEREZ, PASCAL ROMON & ANTONIO ROS

If G is a point group (i. e. G is finite) then, as balls solve the (usual) Euclideanisoperimetric problem, the solutions of the G-isoperimetric problem are balls cen-tered at the fixed points of G. If G is a line group (i. e. if the orbits of G consist ofinfinitely many points lying in a solid cylinder), then it can be showed, see Theorem6, that any G-isoperimetric region is a G-invariant family of either balls or solidcylinders. Thus, in the nonperiodic and the singly periodic cases the problem is wellunderstood. Outside of these two cases we have only partial results, even when Gis a lattice.

If G is neither a point group nor a line group, then the subgroup of translations# = #(G) of G is a lattice of rank two or three and G/# is a finite group. Theseoptions correspond to the doubly periodic and triply periodic cases. In the first one,G is a (3-dimensional) plane group and the quotient space R3/# is the product T 2"Rof a flat 2-torus with the real line. If rank (#) = 3, then G is a space group andthe action of G can be described in terms of the finite group G/# acting on the flat3-torus T 3 = R3/#. Hence, in all the cases, the G-isoperimetric problem in R3, Gbeing a discrete group, is equivalent to the G!-isoperimetric problem on a completeorientable flat three manifold M for a finite group G!. In most parts of this paperit will be more convenient to formulate our problem in this alternative way.

In sections 2 and 3 we review several results on the usual isoperimetric problemon 3-manifolds which can be adapted to the G-invariant context. In particular, wegive a complete solution for singly periodic regions.

In section 4 we study the case M = T (!, ")"R, where T (!, ") is the flat 2-toruswhose lattice is generated by (1, 0) and (!, "), with !2 + "2 # 1 and 0 $ ! $ 1

2(any flat 2-torus is homothetic to just one of these tori). We prove in Theorem 13that, given a finite group G % Sym(M), the area A and the enclosed volume Vof a G-isoperimetric surface of Euler characteristic # < 0 verify the inequality9A3 > 64$|#|V 2. If G = {1}, the natural conjecture says that any isoperimet-ric surface is either a sphere, a cylinder or a pair of parallel planar tori. We provethis conjecture for " # 9!

16 & 1.767 and we show that spheres are isoperimetric (in

their admissible range) for any (!, "), outside a small neighborhood of (1/2,'

3/2)(which corresponds to the hexagonal 2-torus). It is worth noticing that for thehexagonal 2-torus, the conjecture is very close to failing: we have constructed al-ternative candidates (which approximate constant mean curvature surfaces of genustwo) whose area is bigger that the area of the best among spheres, cylinders andplanes enclosing the same volume only by a factor less than 1.0003.

The triply periodic case is the subject of section 5. We focus on the cubic torusT 3 = R3/Z3. In this space we give an isoperimetric inequality for G-invariant re-gions, G being a group with a fixed point and containing the diagonal rotations ofangle 2$/3 (a typical example is the group G = Pm3m). In particular we provethat any surface dividing T 3 in two G-invariant regions of the same volume has arealarger than or equal to 2.19 . The conjectured minimizing surface for this prob-lem is the classical Primitive Schwarz minimal surface whose area is approximately

THE PERIODIC ISOPERIMETRIC PROBLEM 3

Figure 1. P Schwarz minimal surface is conjectured to have leastarea among surfaces dividing the 3-space in two Pm3m-invariant re-gions with equal volume fractions.

2.34 , see Figure 1. We also show that for volumes between 0 and 1/6, the uniqueG-isoperimetric surfaces are round spheres. If we forget the symmetry assumption,then the solution, in the equal volumes case, is known to be a pair a planar toriwith total area 2, see [6, 15, 34]. Although for other prescribed volumes the prob-lem remains open (it is conjectured that any isoperimetric surface in T 3 is either asphere or a cylinder or a pair of planar tori), we prove that the only isoperimetricregions in T 3 which are invariant under the group G are round balls. This excludesa very natural family of alternative candidates of Schwarz type and should be usefulto solve the isoperimetric problem in the cubic 3-torus.

The authors would like to thank Brian White for providing a proof of Theorem 3.

2. Isoperimetry and symmetries in flat three manifolds

Among the global properties of constant mean curvature surfaces in Euclideangeometry, we will need the following ones.

Theorem 1. Let $ be a connected properly embedded surface with constant meancurvature in R3.

• (Alexandrov [2]) If $ is compact, then it must be a round sphere.• (Korevaar, Kusner & Solomon [21]) If $ lies inside a cylinder, then it must be

either a cylinder or a Delaunay unduloid.• (Meeks [27],[21]) If $ is a topological plane (resp. annulus) then it is a plane

(resp. a cylinder or a Delaunay unduloid).

In particular, we have that a compact surface of genus 1 embedded with constantmean curvature in a complete flat three-manifold is either planar or a flat cylinderor an unduloid. The above theorem is proved by using the Alexandrov moving planetechnique. With the same idea we obtain the following standard properties of doubly


periodic constant mean curvature surfaces, see for instance [33]. For convenience westate the result in the quotient space, i. e. the product of a flat 2-torus and a realline.

Proposition 2. If $ is a compact connected constant mean curvature surface em-bedded in T 2 "R with genus g # 2, then $ is symmetric with respect to a horizontaltorus {z = a} and $+ = $({z > a} is a graph over a domain D % T , where T )Dconsists of g)1 closed pairwise disjoint topological disks. In particular, two surfacesof this type with the same symmetry torus (but not necessarily with the same meancurvature) must intersect.

Given a three dimensional Riemannian manifold M3 and a finite group of sym-metries G % Sym(M), we can consider the isoperimetric problem for G-invariantregions (or simply G-isoperimetric problem): among G-invariant regions with pre-scribed volume in M , find those whose boundary has least area. This extends theusual isoperimetric problem (which corresponds to the case G = {1}). In this paperwe will study the situation where G is a finite group of symmetries of a completeorientable flat 3-manifold M (for details about these groups see for instance [16, 22]).

The fundamental existence and regularity properties, see Almgren [4], Gonzalez,Massari and Tamanini [11] and Morgan [28], work in our setting and we can statethe following result.

Theorem 3. Let G % Sym(M) be a finite group of isometries of a compact 3-manifold M . Then for any 0 < v < V (M) there exists a region " % M suchthat

1. g(") = " for any g * G and V (") = v,2. $ = %" minimizes the area among boundaries of all regions satisfying 1.

Any region " satisfying 1 and 2 is compact and has smooth boundary $ with constantmean curvature.

If M is noncompact, then the assertions above still hold, provided that M/Sym(M)is compact.

Proof. We sketch a proof due to Brian White. Given x * M , we denote by Gx thegroup of orthogonal tranformations in R3 (viewed as the tangent space of M at x)obtained by linearizing the stabilizer group {g * G | g(x) = x} of x. The existence of" and regularity of $ outside X = {x * $ : Gx += {1}} hold by the same argumentsused in the usual isoperimetric problem, see [28].

To prove that regularity does hold everywhere consider a point x * X (note that Xconsists of a finite union of points, curves and surfaces). Observe that MonotonicityTheorem and existence of tangent cone holds at x. Indeed, using geodesic normalcoordinates at x, we see that the radial deformations one uses to prove monotonicitypreserve the right symmetries.


The tangent cone C at x is a cone in R3 with vertex at the origin that separates thespace in two Gx-invariant (not necessarily connected) regions "i, i = 1, 2. MoreoverC is area-minimizing among surfaces that separate R3 in two Gx-invariant regions.The cone is a finite union of planar pieces which meet along some half-lines R leavingthe origin. If y += 0 lies in one of these rays, y * R, then after interchanging "1 and"2 if necessary, we have that, for & small enough, each component W of B"(y)("1

is a wedge with interior angle smaller than $. Clearly, the union of the two halfdiscs in %W can be substituted by a surface in W with the same boundary curveand less area. By repeating this construction at all the points of the Gx-orbit of ywe obtain a new surface C ! which contradicts the least area property of C.

Therefore the tangent cone C is a plane, which implies that Gx is either a cyclicor dihedral group which fixes all the points in a geodesic L passing trough x andis orthogonal to C (if Gx is not of this type, we get that x /* $). Then Allard’sregularity theorem [3] says that the surface $ is smooth at x. More precisely: wedo not know a priori that we can apply Allard regularity at the point x, but wecan apply it in all small balls near x that do not intersect X. That shows that,near x, the surface is smooth (which we already knew) and that the tangent planesare nearly perpendicular to L. This means the surface is C1 (or even C1,#) at xand standard removal of singularities theorems in PDE now gives higher regularityaround that point.

A region " satisfying items 1 and 2 in the theorem above will be called a G-isoperimetric region and its boundary $ a G-isoperimetric surface.

If G is a finite group and " is a G-isoperimetric region in a complete orientableflat manifold M , then its boundary $ must be a closed (not necessarily connected)orientable surface with constant mean curvature H . Consider on $ the quadraticform coming from the second variation formula for the area,

Q(u, u) =

!

!

"|,u|2 ) |'|2u2

#dA,(1)

where u lies in the Sobolev space H1($) and ' is the second fundamental form ofthe immersion. As $ minimizes the area up to second order under symmetry andvolume constraints, we have the following useful stability property.

Proposition 4. ([5]). Q(u, u) # 0 for any G-invariant function u * H1($) suchthat

$! u dA = 0.

The function u corresponds to an infinitesimal G-invariant normal deformation ofthe surface, and the zero mean value condition means that the deformation preservesthe volume infinitesimally. The Jacobi operator L = %+ |'|2 is related to Q by theformula Q(u, v) = )

$! uLv dA.

Some of the known properties of the standard isoperimetric problem in 3-manifoldscan be adapted to the G-invariant setting. In particular we have the following result.


Theorem 5. Let M be a complete orientable flat 3-manifold, G a finite group ofsymmetries of M and " % M a G-isoperimetric region bounded by a closed sur-face $ = %". Then, either $ is planar (but not necessarily connected) or $/G isconnected (in particular, the components of $ are all congruent).

If the connected components of $ have genus g = 0 then they are round spheres.If the connected components of $ have genus g = 1 then they must be flat. In

particular, they are tori obtained as a quotient of either a plane or a circular cylinder.

Proof. If $/G is not connected, then there are on $ nonzero locally constant G-invariant functions with zero mean value and the stability condition in Proposition4 implies that $ is totally geodesic.

If the components of $ are topological spheres, then we can lift them to R3 andusing Theorem 1 we conclude that $ is a G-invariant family of round spheres.

If the components of $ are nonflat tori, then Ritore and Ros [32] find a contra-diction for the case G = {1}: they give a function u with mean value zero and suchthat Q(u, u) < 0. In the symmetric setting the function u has the required extrasymmetries and, so the same argument works.

As a direct extension of Theorem 9 of Ritore and Ros [32], using the result abovewe can describe completely the solutions of the symmetric isoperimetric problem inthe singly periodic case. If G is a line group (i. e. the orbits of G in R3 consist ofinfinitely many points lying in a solid cylinder), then there is an infinite cyclic normalsubgroup C % G (the generator of C is either a translation or a screw motion) suchthat G/C is a finite group of isometries of the flat manifold R3/C.

Theorem 6. If G is a line group, then the solutions of the G-isoperimetric problemare spheres (if the prescribed volume V in R3/G is smaller than or equal to a constantV0 = V0(G)) and cylinders (if V # V0).

Proof. Let $ % R3 be a G-isoperimetric surface. As the planar case is clearlyimpossible, from Theorem 5 we have that $/G must be compact and connected.In particular the connected components of the pullback image of $ are properlyembedded constant mean curvature surfaces in R3 which lie inside a cylinder. Sothese components are either spheres, unduloids or cylinders by Theorem 1. From theproof of Theorem 5, we know that unduloids are G-unstable. Finally observe that aunique cylinder around a closed fixed line of G is always better for the isoperimetricproblem that several cylinders parallel to that line. Thus G-isoperimetric surfacesare either a G-invariant family of spheres or circular cylinders and the theoremfollows easily.

If the group G does not contain orientation reversing symmetries, we have thefollowing interesting restrictions.

Theorem 7. Let M be a complete orientable flat 3-manifold, G a finite group of ori-entation preserving symmetries of M , and " % M a G-isoperimetric region bounded


by a closed surface $ = %" with constant mean curvature H. Then either $ isa union of parallel planar 2-tori or $/G is a connected Riemann surface of genusg $ 4 satisfying the following restrictions.

1. If g = 4 then $ is a minimal surface.2. If g = 2 or 3, then A($)H2 $ 2$|G|, where |G| denotes the order of G.

The proof depend on an argument first used by Hersch [17] and Yang and Yau [40],see [31, 41] and [34]. One consider a nonconstant meromorphic map ( : $/G - S2 ofdegree as small as possible. It can be shown that, after composition with a suitableconformal transformation of the 2-sphere, the (G-invariant) coordinate functions of

the map $ - $/G$- S2(1) % R3 are L2-orthogonal to 1. One conclude by using

these coordinate functions as test functions in the stability quadratic form (1).Finally we want to mention some interesting facts (even though we will not use

them in this paper). Ross [35] has proved that the Primitive (resp. Diamond)Schwarz minimal surface of genus three is stable for the isoperimetric problem (i.e.its area is a local minimum among surfaces which enclose a fixed volume) in theprimitive (resp. face-centered) cubic 3-torus.

If # is a lattice generated by two or three orthogonal vectors, then any isoperi-metric surface in M = R3/# admits three pairwise orthogonal mirror symmetries,see [18, 30].

Recently, the double bubble problem in flat 2-tori have been solved, [10]. Forsome partial results in the 3-dimensional case see [9].

3. The isoperimetric profile

The goal of this section is to study a certain di!erential inequality satisfied bythe isoperimetric profile of a complete orientable flat manifold M with a given finitesymmetry group G.

The isoperimetric profile IG = IM,G : (0, V (M)) - R is defined as

IG(v) = inf {A(%") : " % M is a G)invariant region with V (") = v}.In the case G = {1} we will denote the profile simply by I = IM . If the volumeof M is finite then the complementary region of a G-isoperimetric region is a G-isoperimetric region too, and so we have IG(V (M) ) v) = IG(v), for any v.

If " is a region in M bounded by a compact surface $, then we denote by "t and$t = %"t the parallel deformation of $ and ", |t| < &, defined by "0 = ", "t ={p * M | distM(p,") $ t} for t > 0 and "t = {p * " | distM(p,$) # |t|} if t < 0.Note that the regions "t are G-invariant. Then with the notation A(t) = A($t) andV (t) = V ("t) we have the well-known formulas

%&

'

V (t) = v + tA($) + t2HA($) + 2!3 #($)t3

A(t) = A($) + 2tHA($) + 2$#($)t2(2)


where #($) = 12!

$KdA is the Euler characteristic of $. Note that (2) implies

V (t) = V (t0) +$ t

t0A(s)ds.

Consider the following family of ordinary di!erential equations, parametrized bya constant #:

I2I !! + I(I !)2 ) 4$# = 0 ())

Let us look for solutions I(v) of ()) by making a change of variable v = v(t)with the following hypothesis I(v) = dv/dt. Such an assumption leads to a muchsimplified ODE: d3v/dt3 = 4$#. Clearly we deduce that positive solutions of ()) aregiven by the graph of the curves *(t) = (v(t), I(v(t))), with I(v(t)) # 0 and

%&

'

v(t) = V0 + A0t + H0A0t2 + 2!%3 t3

I(v(t)) = A0 + 2H0A0t + 2$#t2(3)

where V0, A0 and H0 are real parameters. These expressions coincide with the onesin formula (2). Note that 2H0 = dI/dv at t = 0 gives the initial slope of the solution.

In the case # < 0, it follows from (3) that for V0, A0 > 0 the function I(v)is a concave function which meets the v-axis orthogonally and has just a singlemaximum. Moreover I is symmetric with respect to the vertical line through themaximum, see Figure 2.

Figure 2. Positive solutions of ()) for di!erent values of the para-meter #.

Recall that given an ODE of the form F (u!!, u!, u) = 0, with %F/%u!! > 0, anda continuous function w(t) defined in an open interval, we say that w satisfies thedi!erential inequality F (w!!, w!, w) $ 0 (in the weak sense) at t = t0 if there isa solution u(t) of the ODE defined for |t ) t0| < &, such that w(t0) = u(t0) andw(t) $ u(t). We will say that v .- J(v), a < v < b, is a supersolution of ()) if it iscontinuous and satisfies J2J !! + J(J !)2 ) 4$# $ 0 (in the weak sense) at each valueof v. Note that if I is a solution of ()) with constant #!, then I is a supersolution


of ()) for any constant # # #!. Throughout this paper we will use several times thefollowing versions of the maximum principle.

Lemma 8. Let I : [v1, v2] - R be a positive solution of ()) with constant # andJ : [v1, v2] - R a positive continuous function which is a supersolution of ()) in]v1, v2[ with constant #!. Assume #! $ min {0,#}.

i) If I ) J has a nonnegative maximum at the interior of the interval, then I = Jin the whole interval. In particular, if I)J is positive and monotonically increasingnear v1, then the same holds in [v1, v2].

ii) If I and J coincide at the values v1 and v2, then either I < J on open interval(v1, v2) or both functions coincide in that interval.

iii) If J # I in [v1, v2] and I(v0) = J(v0), for some v0 with v1 < v0 < v2, thenI = J in [v1, v2].

iv) If I(v1) = J(v1) and J !+(v1) = lim supv"v+

1(J(v) ) J(v1))/(v ) v1) is smaller

than or equal to I !(v1), then I # J . Moreover I(v0) = J(v0) at some interior valuev0, implies that I = J in [v1, v0].

Proof. To prove i), suppose that I)J attains its maximum at the interior point v and(I)J)(v) # 0. We can also assume that J is not only a supersolution but a solutionof ()) with constant #!. Hence, J(v) $ I(v), J !(v) = I !(v) and J !!(v) # I !!(v). If# > #!, then using ()) we obtain

J !!(v) +(J !)2(v)

J(v)=

4$#!

J(v)2<

4$#

I(v)2= I !!(v) +

(I !)2(v)

I(v)$ J !!(v) +

(J !)2(v)

J(v)

and this contradiction proves the claim. The same argument works in the case# = #! < 0 and J(v) < I(v). If # = #! and J(v) = I(v), the result follows fromthe uniqueness of solutions of ()) with given initial data. Finally, if # = #! = 0 theproperty can be checked directly (note in this case I and J are functions of the typev .-

'av + b).

Items ii) and iii) follow directly from the first one.To prove iv) choose another solution I+ of ()) with constant #, starting at the

same point (v1, I(v1)) than I and with a slightly greater slope. So that for v closeto and greater than v1, I+ lies strictly above J . Item i) says that I+ ) J does notadmit an interior positive local maximum and, therefore it must be monotonicallyincreasing, first near v1 and then in the whole (v1, v2]. As I+ can be take abitrarilyclose to I, we have that I # J in [v1, v2] and (I ) J)(v) $ (I ) J)(w) for v < w.

Theorem 9. Let " be a G-isoperimetric region with V (") = v in a complete flat3-manifold M and denote by H the mean curvature of $ = %" (with respect to theinwards pointing unit normal vector field).


a) The isoperimetric profile IG is continuous, has left and right derivatives I !G+(v)

and I !G!(v), for any 0 < v < V (M), and

I !G+(v) $ 2H $ I !

G!(v).(4)

Moreover there are G-isoperimetric regions "+ and "# whose mean curvatures aregiven by I !

G+(v)/2 and I !G!(v)/2 respectively. In particular, IG is di!erentiable at v

provided that there exists a unique isoperimetric surface enclosing a volume v.

b) For 0 < v < V (M), IG satisfies (in the weak sense) the di!erential inequality

I2GI !!

G + IG(I !G)2 ) 4$#($) $ 0,(5)

where #($) is the Euler characteristic of $ (in particular, IG is concave). If IG istwice di!erentiable in a neighborhood (v1, v2) of v, then I2

GI !!G +IG(I !

G)2)4$#($) = 0in (v1, v2) if and only if any isoperimetric surface enclosing a volume between v1 andv2 is spherical, cylindrical or planar (this holds, in particular, for $ itself).

c) Let k be the minimum number of points in the orbits of G. Then, for small v,any G-isoperimetric region of volume v is a G-invariant family of k metric balls. Inparticular

IG(v) = (36k$v2)1/3, for v close to 0.(6)

Proof. The first part of a) if proved as in the nonsymmetric case, see [7]. Theexistence of the regions "± and the statement c) follow from compactness arguments,see Theorem 2.2 in Morgan and Johnson [29] and Theorem 18 and Proposition 5 in[34].

The di!erential inequality (5) is a restatement (working only in the three dimen-sional case) of a result of Bavard and Pansu [7]. To prove it, we consider the paralleldeformation $t of $. Using (2) we have that V !(t) = A(t) and writing area in termsof volume we get

dA

dV=

A!(t)

V !(t)=

A!(t)

A(t)and

d2A

dV 2=

A!!(t)A(t) ) A!(t)2

A(t)3.

Therefore

A2 d2A

dV 2+ A

(dA

dV

)2

= A!!(t) = 4$#($).

Moreover IG(v) = A(v) and, by the minimizing property of the profile, IG(V ) $A(V ) for V near v which proves (5).

The concavity of the profile is clear at the points v where #($) $ 0. At the pointswhere #($) > 0, Theorem 5 says that isoperimetric surfaces are spheres and theconcavity can be checked directly.

To prove the remaining part of b), suppose that

I2GI !!

G + IG(I !G)2 ) 4$#($) = 0 in (v1, v2).(7)


Observe that both functions IG(V ) and A(V ) are solutions of (7) with IG(v) = A(v)and I !

G(v) = A!(v). Therefore, IG and A coincide in a neighborhood of v. Thus theparallel surfaces $t to $ are isoperimetric surfaces, and so they all have constantmean curvature, which is only possible when $ is spherical, cylindrical or planar. Inparticular, #($) # 0. Now consider an isoperimetric surface $! enclosing a volumeV * (v1, v2). Then, we have

4$#($!) # I2G(V )I !!

G(V ) + IG(V )(I !G(V ))2 = 4$#($) # 0.

Therefore the connected components of $! have genus 0 or 1 and we conclude byusing Theorem 5 that these components are spherical, cylindrical or planar.

Finally, since parallel deformation takes spheres to spheres, equality (7) holds in(v1, v2) whenever spheres are isoperimetric in this range. The same holds cylindersand planes, and we have proved b).

A closed constant mean curvature G-invariant surface $ enclosing a domain inM is said to be G-stable if the quadratic form (1) verifies Q(u, u) # 0 for any G-invariant function u with

$! u dA = 0. To finish this Section, we will prove that the

di!erential inequality (5) holds for 1-parameter families of G-stable surfaces.

Proposition 10. Let M be a complete orientable flat 3-manifold, G % Sym(M) afinite group and {$v | v1 < v < v2} a smooth family of G-stable surfaces in M , suchthat the volume enclosed by $v is equal to v. Then, the area function A(v) = A($v)satisfies the di!erential inequality

A2A!! + A(A!)2 ) 4$# $ 0,

where # is the Euler characteristic of $v (which does not depend on v).

Proof. For fixed volume v, denote by $ = $v, u : $v - R the infinitesimal normalvariation of the family at $, and L the Jacobi operator of $. Standard variationformulae for the volume, area and mean curvature H give

$! u = )1, A!(v) =

)2H$! u = 2H and A!!(v) = 2H !(v) = Lu.

Now define a = )1/A. As $ is G-stable and u) a is a G-invariant function withzero mean value, we have

0 $ Q(u ) a, u ) a) = Q(u, u) ) 2Q(u, a) + Q(a, a)

= )!

!

uLu + 2a

!

!

Lu ) a2

!

!

|'|2 =

()

!

!

u + 2aA

)A!! ) 1

A2

!

!

(4H2 ) 2K)

= )A!! ) (A!)2

A+

4$#

A2,

where K denotes the Gauss curvature of $ and we have used the Gauss-BonnetTheorem.


4. The doubly periodic case

Any flat 2-torus T 2 is homothetic to a torus T (!, ") = R2/#(!, ") defined bythe lattice #(!, ") spanned by the vectors (1, 0) and (!, "), with !2 + "2 # 1,0 $ ! $ 1/2 and 0 < ". The area of T (!, ") and the length of its shortest closedgeodesic are given by " and 1, respectively. In particular

A(T (!, ")) #'

3

2,(8)

with equality for the hexagonal torus, i. e. the quotient of R2 over the hexagonallattice generated by two unit vectors forming an angle of $/3.

The conjectured candidates for isoperimetric surfaces in T (!, ")"R are spheres,cylinders and pairs of horizontal planes. Namely, for 0 < r < 1/2 the round sphereof radius r embeds isometrically in T (!, ") " R, and if we write area as functionof volume we have A(V ) = (36$V 2)1/3. Analogously, the best cylinders for theisoperimetric problem are those constructed around a closed geodesic of length 1.If the cylinder has radius r, 0 < r <

'3/4, then its area A(r) = 2$r and volume

V (r) = $r2 are related by the equation A(V ) = 2'$V , while the area of a pair of

parallel planar tori equals A(V ) = 2".We define the spheres-cylinders-planes profile (scp profile) of T (!, ") " R as the

function Iscp : (0,/) - R+ which gives the least area among spheres, cylinders andpairs of parallel planes enclosing a volume V . From the paragraph above we havethat (see Figure 3)

Iscp(V ) =

%****&

****'

(36$V 2)1/3 if 0 < V $ 4!81 spherical,

2'$V if 4!

81 $ V $ &2

! cylindrical and

2" if &2

! $ V planar range.

Figure 3. The spheres-cylinders-planes profile in T (!, ")" R.


Recall that 2H is the is the slope of Iscp at its smooth points. The relationshipbetween area, volume and mean curvature is H = A

3V in the spherical branch,H = A

4V in the cylindrical range and H = 0 in the planar range.If we consider the isoperimetric problem in T 2 " R with no extra symmetries, it

is an interesting and natural problem to decide for which tori T (!, ") the scp profilecoincide with isoperimetric profile. Consider the following statement.

Conjecture 1. The unique solutions of the isoperimetric problem in T (!, ") " Rare spheres, cylinders around closed geodesic and parallel pairs of totally geodesictori. In particular, the isoperimetric profile of T (!, ") " R coincides with its scpprofile.

Using compactness arguments, Ritore and Ros [32] proved that the conjectureholds for " large enough. We will now give an explicit upper bound on ".

Let $ % T 2 " R be a compact, connected, embedded surface with Euler char-acteristic #, constant mean curvature H , area A and enclosed volume V . Denoteby N = (N1, N2, N3) the inwards pointing unit normal vector along $. We shallconsider a family of deformations of $ in T 2 " R, obtained by applying the a&netransformation (' : (x1, x2, x3) .- (x1, x2, (1 + +)x3). Henceforth O(tn) will denotea function such that O(tn)/tn is bounded when t - 0.

Lemma 11. If $ is a compact surface with constant mean curvature embedded inT 2 " R, then the area of the surface $' = ('($), with |+| < &, is

A($') = A + +

!

!

(1 ) N23 ) dA +

+2

2

!

!

N23 (1 ) N2

3 ) dA + O(+3),

and the volume enclosed by $' is (1 + +)V .

Proof. In order to compute the variation of the area, it is enough to consider a pieceS of $ given by the graph of a function (x, y, u(x, y)), with (x, y) * D. For thatsurface we have

N3 =±1+

1 + |,u|2, A(+) = A(S') =

!

D

+1 + (1 + +)2|,u|2 dxdy,

and therefore

A!(0) =

!

D

|,u|2

1 + |,u|2+

1 + |,u|2 dxdy =

!

S

(1 ) N23 ) dA

and

A!!(0) =

!

D

|,u|2

(1 + |,u|2)3/2dxdy =

!

S

N23 (1 ) N2

3 ) dA.


Let G be a finite subgroup of Sym(T 2 "R). We will assume that G fixes the planartorus x3 = 0. By applying the transformation (x1, x2, x3) .- (x1, x2, (1 + +)x3) tothe parallel surface $t of $ we have a two-parameters deformation of G-invariantsurfaces (t,+) .- $t,'. This deformation allows us to obtain the following usefulinformation.

Proposition 12. Let $ % T 2"R be a nonplanar G-isoperimetric surface with Eulercharacteristic #, mean curvature H, area A and enclosed volume V . Then,

!

!

N23 dA = A ) 2HV(9)

and

4HV ) 12H2V 2

A+

4V 2

A2$#+

!

!

N23 (1 ) N2

3 ) dA # 0,(10)

where N = (N1, N2, N3) is the Gauss map of $.

Proof. The area and volume functions of the parallel surfaces t .- $t are

A(t) = A($t) = A + 2tHA + 2$#t2 , V (t) = V + tA + t2HA + O(t3).(11)

Therefore, Lemma 11 and the fact that the surfaces $ and $t have the same normalvector, imply that the area and volume functions of (t,+) .- $t,' are

A(t,+) = A($t,') = A(t) + +

!

!

(1 ) N23 ) dA(t) +

1

2+2

!

!

N23 (1 ) N2

3 ) dA(t) + O(+3)

and

V (t,+) = (1 + +)V (t),

where dA(t) = (1 + 2tH + t2K)dA stands for the area element of $t. If we taket = t(+), expanding t in a neighborhood of + = 0 as

t(+) = t1++t22+2 + O(+3),(12)

a straightforward computation gives that the area function of the composed defor-mation is

A(+) = A(t(+),+) = A + +

(2AHt1 +

!

!

(1 ) N23 ) dA

)

+ +2

,2$#t21 + AHt2 + 2Ht1

!

!

(1 ) N23 ) dA +

1

2

!

!

N23 (1 ) N2

3 ) dA

-+ O(+3).

Analogously, the volume function is

V (+) = (1 + +)V (t(+)) = V + +(V + At1) + +2

,AHt21 + At1 +

A

2t2

-+ O(+3).

(13)


From now on, we choose t(+) such that the deformation keeps the volume fixed. By(13), this is implies

t1 = )V

Aand t2 =

2V

A) 2HV 2

A2.

Plugging these equalities into the area function it follows that

A(+) = A + +

()2HV +

!

!

(1 ) N23 ) dA

)+ +2

,+2$#

V 2

A2+ 2HV

)2H2V 2

A) 2HV

A

!

!

(1 ) N23 ) dA +

1

2

!

!

N23 (1 ) N2

3 ) dA

-+ O(+3).

The constant mean curvature assumption means exactly that the coe&cient of +vanishes, which gives the first formula in the statement. The stability of $ underG-invariant volume preserving deformations implies that A!!(0) # 0, and the secondformula in the Proposition follows after substitution of

$!(1)N2

3 ) dA by 2HV .

Theorem 13. Let G be a finite group of symmetries of T 2 " R and $ a nonplanarG-isoperimetric surface with Euler characteristic # < 0, mean curvature H, area Aand enclosed volume V . Then,

64$

9|#| <

A3

V 2and

3A

16V

.

1 )/

1 + #64$V 2

9A3

0

< H <3A

16V

.

1 +

/1 + #

64$V 2

9A3

0

Proof. Using Cauchy-Schwarz inequality and the equality in Proposition 12, we canestimate!

!

N23 (1 ) N2

3 ) dA =

!

!

N23 dA )

!

!

N43 dA

$!

!

N23 dA ) 1

A

(!

!

N23 dA

)2

= 2HV ) 4H2V 2

A.

Clearly, the inequality above must be strict, otherwise N3 would be constant, whichcontradicts the hypothesis.

Joining the above inequality with the one in Proposition 12 we obtain

0 < 3H ) 8H2V

A+

2V

A2$#.(14)

The right-hand-side of (14) can be considered as a parabola in H with negativemain coe&cient, thus its discriminant must be positive, i. e. 9A3 + 64$V 2# > 0.Moreover, H must be trapped between the two roots of 3H ) 8H2V

A + 2VA2$# = 0.

Now the Theorem is proved.


Corollary 14. In T 2 "R, any G-isoperimetric surface with genus greater than oneencloses a volume strictly less than 3

4$

!A(T )3/2.

Proof. Use the isoperimetric inequality in Theorem 13 together with A $ 2A(T ).

As a consequence of Theorem 13, we can solve Conjecture 1 for a large family ofspaces of the form T 2 " R. We denote by

A = {(!, ")| T (!, ") " R does not satisfy Conjecture 1 }.Corollary 15. If " # 9!

16 & 1.767, then T (!, ")"R satisfies Conjecture 1, i. e. theunique isoperimetric surfaces in T (!, ")" R are spheres, cylinders and planes.

Proof. As Conjecture 1 is true for large ", see [32], to prove the proposition it isenough to see that any pair (!, ") in %A satisfies " < 9$/16. The compactness of thespace of isoperimetric surfaces implies that for (!, ") * %A the isoperimetric profile Iof T (!, ")"R coincides with its scp profile, but there exists an isoperimetric surface$ % T (!, ")"R other than spheres, cylinders and planar tori. From Theorem 5 weknow that $ has Euler characteristic # $ )2. In particular, if A and V denote thearea of $ and the volume it encloses, then we have A = I(V ). On the other hand,Theorem 9 implies

I2I !! + I(I !)2 ) 4$# $ 0 at V,(15)

which is impossible if V lies at the interior of either the spherical range or thecylindrical or the planar one by item b) of Theorem 9. Thus V lies at one of thetransition points of I, that is either (V, A) = (4$/81, 4$/9) or (V, A) = ("2/$, 2").

In the first case V separates the spherical and the cylindrical branches, and themean curvature H of $ must be greater than or equal to the one of the cylinder withthe same area (and enclosing the same volume): to see that observe that, otherwise,for & > 0 su&ciently small, the equidistant surfaces $t, 0 < t < &, would have lessarea than the cylinders enclosing the same volume. Hence

H2A # (A

4V)2A =

9$

4> 2$,

which contradicts Theorem 7. Therefore $ has area A = 2" and encloses a volumeV = "2/$. Plugging these values in the first inequality of Theorem 13 we obtain" < 9$/16.

Remark 1. From arguments in the proof of the last Corollary one deduces that if(!, ") * %A and $ % T (!, ")"R is an isoperimetric surface with Euler characteristic# $ )2, then the volume enclosed by $ separates the spherical and cylindricalbranches of the profile. After substitution of the values V = &2

! , A = 2" in thesecond inequality of Theorem 13, we obtain the following additional informationabout the nonstandard isoperimetric surface $:

3$

8"

.1 )

/1 + #

8"

9$

0< H <

3$

8"

.1 +

/1 + #

8"

9$

0.


Proposition 16. If A += 0, then the boundary %A is a graph over its horizontalprojection (!, ") .- ! on [0, 1/2] and A lies below this graph.

Proof. Consider (!, ") * %A and suppose that " >'

1 ) !2. As A is compact, toprove the Proposition it is enough to see that, for some , < ", {(!, " !) | , < " ! < "}is contained in the interior of A.

The isoperimetric profile of T (!, ") " R coincides with its scp profile and thereexists an isoperimetric surface $ % T (!, ") " R with genus g # 2. By the proof

of Corollary 15 we know that $ encloses a volume V = &2

! and has area A = 2".We shall consider a deformation of $, composed of the deformation through parallelsurfaces $ .- $t % T (!, ")" R (t close to zero), followed by the a&ne deformation(which changes the ambient space!)

-' : T (!, ") " R - T (!, (1 + +)") " R | -'(x, y, z) = (x, (1 + +)y, z),

where + is close to zero.The area and volume functions A(t,+), V (t,+) of (t,+) .- $t,' = -'($t) are given

by expressions similar to those in the proof of Proposition 12 after replacing N3 withN2 for obvious reasons. Taking t = t(+) = t1++ O(+2), we find

A(+) = A($t('),') = A + +

(2AHt1 +

!

!

(1 ) N22 ) dA

)+ O(+2),

while

V (+) = V (t(+),+) = (1 + +)V ("t(')) = V + +(V + At1) + O(+2).

We shall determine the function t(+) by using the condition that the volume func-tion V (+) of the composed deformation coincides with the volume separating thecylindrical zone and the planar zone in the scp profile Iscp,' for the new ambientspace T (!, (1 + +)") " R, which is given by (1 + +)2V = (1 + +)2"2/$.

This condition is easily seen to be equivalent (at the first order) to the equationt1 = V/A which gives the area function

A(+) = A + +

(2HV +

!

!

(1 ) N22 ) dA

)+ O(+2)

= A + +

(!

!

(1 ) N23 ) dA +

!

!

(1 ) N22 ) dA

)+ O(+2)

= A + +

!

!

(1 + N21 ) dA + O(+2).

where we have used Proposition 12 in the second equality. Let us compare the areaof $t('),' and the scp profile (whose value at (1++)2V is just 2(1++)" = (1++)A).


Order one information is su&cient and we have

A(+) ) Iscp,'((1 + +)2V ) = +

(!

!

(1 + N21 ) dA ) A

)+ O(+2)

= +

!

!

N21 dA + O(+2).

We point out that the last integral,$! N2

1 dA cannot be zero, otherwise N1 vanishesidentically on $, thus $ must be flat which contradicts our assumption on the genusof $. We conclude by choosing + < 0 close to 0 so that (!, (1 + +)") lies in theinterior of A.

4.1. Isoperimetry of spheres. Our next goal is to prove that except for a verysmall range in the moduli space of tori, the only isoperimetric surfaces in T 2 " R(no symmetries are imposed, i. e. G = {1}) on the whole spherical range of thescp profile (volumes less than or equal to 4$/81) are spheres. In particular, thisproperty will be true for any rectangular torus. To obtain such result we will needthe following inequality.

Proposition 17. Let $ be a compact, connected surface with genus g # 2 in T 2"Rwith constant mean curvature H, area A and enclosing a volume V . Then,

AH2(A ) 2HV ) + 2$+

A(A ) 2HV ) # 4$A(T 2).

Proof. From Proposition 2 we know that $ is symmetric respect to a horizontalplane (which can be supposed to be x3 = 0) and that $+ = $ ( {x3 # 0} is thegraph of a function defined on a domain D % T , whose boundary %D consists of g)1Jordan curves. As these Jordan curves bound disks in T , the euclidean isoperimetricinequality applied in T ) D gives

L(%D)2 # 4$(A(T ) ) A(D)),(16)

where as usual, A, L denote area and length of the corresponding objects. The Jaco-bian of the projection of $+ onto D is given by )N3, N3 being the third componentof the inwards pointing unit normal vector along $. Hence A(D) = )

$!+ N3 dA.

On the other hand, applying the divergence Theorem to the vector field X = tangentpart of e3 on $+ (whose divergence is 2HN3), we obtain

)2HA(D) = 2H

!

!+

N3 dA =

!

(!+

1X,)e32 ds = )L(%D),

which combined with (16) implies

H2A(D)2 + $A(D) ) $A(T ) # 0.(17)

From Schwarz inequality one has

A(D)2 =

(!

!+

N3 dA

)2

$ A

2

!

!+

N23 dA =

A

4(A ) 2HV ),(18)


where we have used Proposition 12 in the last equality. Combining (17) and (18)the Proposition follows.

Theorem 18. Let "0 = 2!27

13 )

'2 +

+9 ) 3

'22. For any " > "0, the unique

isoperimetric surfaces $ % T (!, ") " R with enclosed volume 0 < V $ 4!81 are

spheres and cylinders (the last ones only for V = 4!81 ).

Remark 2. To give a idea of how restrictive the condition " > "0 is, note that"0 & 0.876603 and the minimum value of " representing tori T (!, ") is " =

$3

2 &0.866025. In particular, Theorem 18 applies to all rectangular tori (i. e. ! = 0 and" # 1).

Proof. Suppose that for a given T = T (!, "), there exists an isoperimetric surfacein T " R with genus g # 2 enclosing a volume V * (0, 4!

81 ]. If "1 is the largest "among tori satisfying this property (Corollary 15 implies that "1 $ 9$/16), thenfrom the maximum principle in Lemma 8 and the compactness of the space ofisoperimetric surfaces it follows that, for some !, there must exist an isoperimetricsurface $ % T (!, "1) " R with genus g # 2, area A = 4!

9 and enclosed volumeV = 4!

81 . Using Proposition 17 with these area and volume we get

2$

729

118H2 ) 4H3 + 27

'9 ) 2H

2# A(T1) = "1,(19)

where H is the mean curvature of $ and T1 = T (!, "1). On the other hand, theinequality AH2 $ 2$ of Theorem 7 ensures that H $ 3$

2.

Define f(h) = 2!729(18h2)4h3+27

'9 ) 2h), with h * [0, 3$

2]. Equation (19) implies

that "1 $ f(H) $ max[0,3/$

2] f . A direct computation shows that f ! has the samesign as the polynomial P (h) = (36h ) 12h2)2(9 ) 2h) ) 272. By the intermediatevalue Theorem it can be checked that P has five real roots, one in each interval()1, 0), (0, 1), (5

2, 3), (3, 4), (4, 5). In particular, there exists h0, 0 < h0 < 3$2, such

that P (h0) = 0, P < 0 in [0, h0) and P > 0 in (h0,3$2] (in fact, h0 & 0.285499). This

implies that f is decreasing in [0, h0), increasing in (h0,3$2] and has a local minimum

at h0. As f(0) < f( 3$2), we deduce that max[0,3/

$2] f = f( 3$

2) = "0, which finishes

the proof.

4.2. Lamellar catenoids. After Remark 1 it is likely that, in order to proveConjecture 1 for T (!, ") " R, the hardest situation holds for the hexagonal torus

T (12 ,

$3

2 ) " R and, in this space, for the volume V = 34! which separates the cylin-

drical and the planar branches. The cylinder (and the pair of planar tori) enclosingthis volume have area equal to

'3. The most serious alternative candidates to solve

the isoperimetric problem in that case are constant mean curvature surfaces $ ofgenus two which are symmetric with respect to the planar torus z = 0 and suchthat $ ( {z = 0} is a convex curve, see Figure 4 (these surfaces are called meshes


or lamellar catenoids in materials science). The first surfaces of this type whereobtained by Lawson [23] by using the conjugate Plateau problem construction, seealso Grosse-Brauckmann [12, 13] and Ritore [30].

In this subsection we will construct a surface which approximates one of the abovesurfaces and we will prove that its area is only slightly greater than the conjecturedvalue, which indicates that the area of the actual constant mean curvature meshis (almost) equal to

'3. This somewhat surprising fact shows that either to prove

or disprove the natural conjecture in T (12 ,

$3

2 ) " R we will need to use very sharpestimates.

Claim. There is a lamellar catenoid shaped surface, with Euler characteristic )2,in T (1

2 ,$

32 ) " R, enclosing a volume V = 3

4! and whose area lies between Iscp(34! ) ='

3 and 1.0003 "'

3.

Figure 4. Left, center: Genus two surfaces with constant meancurvature in the product of the hexagonal torus and the real line.Right: Fundamental pieces which, after either reflexions or transla-tions, fill up the whole surface. Computer graphics by Karsten Grosse-Brauckmann, [13].

Let D be the largest Euclidean disk embedded in the hexagonal torus. Since thistorus admits a regular hexagon as fundamental domain, this disk is the one touchingall the edges of the hexagon and so D has radius 1/2. We will consider connected

piecewise smooth genus two surfaces $ in T (12 ,

$3

2 ) " R obtained as an annulus ofrevolution $1 inside the cylinder D " R (the axis of revolution being the verticalline through the center of D) glued along its boundary with the union of two planar

pieces $0 = (T (12 ,

$3

2 ))D)"{±d}, for some d > 0 (to be determined). We will alsoassume that $ is symmetric with respect to the horizontal torus z = 0. The area of


$ and the volume it encloses are given by

A = A($1) + A($0) = A($1) +'

3 ) $2

and

V = V1 + V0 = V1 + ('

3 ) $2)d,

respectively, where V1 is the volume enclosed by $1 inside the cylinder D " R andV0 is the volume, outside the cylinder, enclosed by $0. In order to choose the bestcandidates to solve the isoperimetric problem, we will assume that $ is a criticalpoint of the area A, for variations in our family fixing the volume V . Then, it is clearthat $1 must be a Delaunay surface, i. e. a surface of revolution with constant meancurvature H (we orient $ by its inwards pointing unit normal vector field N alongthe smooth pieces of the surface). To see what is the correct boundary condition forthis variational problem, we consider an infinitesimal normal variation of $, whichconsists of a revolution invariant function u on $1, symmetric with respect to z = 0,and a constant function , on $0. Moreover, along the boundary of the annulus $1,we have u) , cos! = 0, where ! is angle that the outward pointing unit co-normal. of $1 makes with the horizontal plane z = d. This compatibility condition saysthat the surfaces $1 and $0 remain glued (at least at the infinitesimal level) alongtheir boundaries during the variation.

As V remain fixed, the first variation formula for the volume gives

0 =

!

!1

u dA +1'

3 ) $2

2,.(20)

As we are assuming that $ is a critical point of the area functional for the variationsabove, from the first variation formula for the area we obtain

0 = )2H

!

!1

u dA )!

(!1

, sin! ds = )2H

!

!1

u dA ) 2$, sin!(21)

Finally from (20) and (21) we conclude that

sin! =

.'3

$) 1

2

0H,(22)

which is the boundary condition we wanted.In our context, it is natural to consider H > 0, which corresponds, among De-

launay surfaces, to the case of nodoids. To compute explicitly areas and volume weneed to recall the parametrization of these surfaces in terms of elliptic functions, seefor instance [24]. Consider constants a, b with a < 0 < |a| < 1

2 < b and define thefunctions

m(a, b) = 1 ) a2

b2, ((a, b, r) = arcsin

3b2(r2 ) a2)

r2(b2 ) a2),


and

z(a, b, r) = )aF (((a, b, r), m(a, b)) ) bE(((a, b, r), m(a, b)) +

+(r2 ) a2)(b2 ) r2)

r,

where |a| < r < b and F and E are the elliptic integrals of the first and second kind,given by

F ((, m) =

! $

0

dt+1 ) m sin2 t

and E((, m) =

! $

0

+1 ) m sin2 t dt .

It is also useful to know that, if z! denotes the derivative with respect to r, then

z!(a, b, r) = ) r2 + ab+(r2 ) a2)(b2 ) r2)

.

Nodoids form a two-parameter family, nodoid(a, b), and each of these is defined bythe radial graph ±z(a, b,

+x2 + y2). In our case we want to consider only the piece

$1(a, b) defined by |a| < r < 1/2, see Figure 5.

Figure 5. The nodoid portion inside the lamellar catenoid.

The mean curvature of nodoid(a, b) is H = 1a+b and the relation tan! = z!(a, b, 1

2)gives

sin! =z!+

1 + (z!)2

44444(a,b,1/2)

= ) 1 + 4ab

2(a + b).

Therefore, the boundary condition (22) transforms into b = )$

32!a . So we have a

1-parameter family of lamellar catenoids type surfaces $(a) and, using the notationabove, we have the following formulae for the volume and the area determined bythese surfaces:

V (a) = 4$

! 1/2

#a

rz(r) dr + ('

3 ) $2)z(

1

2) ,


A(a) = 4$

! 1/2

#a

r+

1 + z!(r)2 dr +'

3 ) $2.

Numerical computation give that the equation V (a) = 34! has a solution a0 &

0.2363711254 and for this value the area of the corresponding surface is A(a0) &'3 + 0.00048 < 1.0003 "

'3, as we claimed.

5. Isoperimetric inequalities for triply periodic regions

The purpose of this section is to give an isoperimetric inequality for G-symmetricsurfaces in the cubic torus T = R3/Z3, where G is any finite group of isometries of thetorus fixing the origin of T and containing the diagonal rotations through the origin.These are rotations of angle ±2$/3 around the axes of directions (±1,±1, 1). Wewill eventually show that the isoperimetric profile IG for this problem lies above thespheres-cylinders-planes profile Iscp, which in particular implies that G-symmetricsurfaces in T other than spheres cannot be isoperimetric in T (for the nonsymmetricproblem).

It can checked that the groups G satisfying the restrictions above are (takingpullback groups in R3 and using crystallographic notation, see [22, 16]) P23, P432,P43m, Pm3 and Pm3m. They leave two points fixed, (0, 0, 0) and (1

2 ,12 ,

12), and

the first (resp. the last) one is a subgroup (resp. supergroup) of the others. Weconjecture that all these groups have the same isoperimetric surfaces which (whenviewed in the torus) consist of either a single sphere (for small volumes) or a genusthree surface of Schwarz type (for volumes close to 1/2). In particular, amongsurfaces dividing the torus in two equal volume G-invariant regions, the classicalSchwarz minimal surface should be the one with smaller area, see Figure 1. We willprove in Theorem 20 that this area must be at least 2.19 (the actual area of Schwarzminimal surface is & 2.34). We will also prove that spheres are G-isoperimetric forvolume smaller than 1/6 & 0.17. According to the numerical computations in [1],spheres should be optimal up to volume & 0.29.

Proposition 19. Any solution $ of the isoperimetric problem for G-symmetric sur-faces in the cubic torus T is either a single sphere (centered at a fixed point of G)or has Euler characteristic at most )4.

Proof. The key property to prove the proposition is that G contains rotations ofangle 2$/3 with axes in three independent directions. We can assume that thevolume V of the region " enclosed by $ satisfies 0 < V $ 1/2. From the concavityof the profile IG, see Theorem 9, we have that " is mean convex, i.e. the meancurvature of $ (with respect to the inwards pointing normal) is # 0.

It ought to be noted that either $ is a union of (quotients of) planes or thequotient surface $/G is connected, see Theorem 5, and so the components of $ areall congruent (via symmetries of G) and consist of closed orientable surfaces. In


the first case, embeddedness forces the planes to be parallel to a single fixed plane,which is impossible under the action of G.

If these components have genus zero then, by Theorem 1, they must be roundspheres of the same radius r > 0. As there exists a single sphere centered at a fixedpoint of G (hence G-invariant) enclosing any volume between 0 and 1/2, it is clearthat this sphere will do better than any other nonconnected G-invariant family ofspheres.

Suppose now that the components of $ have genus one. These surfaces consistof union of n (quotients of) cylinders. The orbit under G of a cylinder may yieldan embedded disconnected surface, however it turns out to be nonoptimal: to seethis, note that such cylinder is the tubular neighborhood of radius r > 0 of a closedgeodesic of T of length /, so that the total area of $ is A = 2$n/r and the enclosedvolume V = $n/r2. Since we assume that this surface is isoperimetric, it has todo better than the competitor sphere of the same volume. Using the isoperimetricprofile for spheres, this condition amounts to:

36$ # A3

V 2=

8$3n3/3r3

$2n2/2r4=

8$n/

r.

However / is certainly greater than or equal to 1, while the property of G mentionedabove forces n # 3. We infer that r # 2/3, which gives a volume V # 4$/3 whichis much more than the limit 1/2.

Next consider the case where $ is a connected surface with genus two. We discardthis possibility by considering the full lift 5$ of $ to R3. As the mean convex regionenclosed by $ is a solid donut, see Meeks [26], it follows that the components of5$ are at most doubly periodic. If they were nonperiodic (resp. singly periodic)then, by Theorem 1, they should be spheres (resp. Delaunay surfaces or cylinders).As both options contradict the genus assumption, we conclude that 5$ consists oftranslated images of a certain doubly periodic constant mean curvature surface inR3. In particular, each connected component of 5$ lies in a slab bounded by twoparallel planes to a fixed plane in R3, which is impossible as 5$ is invariant underthe pullback group 5G of G. Hence the contradiction.

As conclusion we have shown that either $ is a (single) sphere or it is a connectedsurface of genus at least three or it has at least two genus two components, the lattercases giving #($) $ )4, as we claimed.

The profile of spheres is given by the function Isph(v) = (36$v2)1/3. We now definea function J : [0, 1] - R by J(v) = Isph(v) if v $ 1

6 , and J is the solution of

J2J !! + J(J !)2 + 16$ = 0(23)

with J(16) = Isph(

16) and J !(1

6) = I !sph(

16), if 1

6 $ v $ 56 , and J(v) = Isph(1 ) v) if

56 $ v $ 1. Note that (23) is the ODE which appears in Theorem 3 correspondingto Euler characteristic )4. The function J is symmetric with respect to v = 1/2


and using (3) we deduce that for 16 $ v $ 5

6 , the graph of J can be parametrized by

v(t) =1

6+ $1/3t + 2$2/3t2 ) 8$

3t3 , a(t) = $1/3 + 4$2/3t ) 8$t2 .

Theorem 20. The G-isoperimetric profile IG for a group of symmetries G of thecubic torus T = R3/Z3 fixing a point and containing the diagonal rotations (ofangle 2$/3) through this point, is bounded below by the function J , IG # J . Inparticular, any G-symmetric surface dividing T in two equal volumes has area largerthan J(1

2) # 2.19 and any G-isoperimetric surface enclosing a volume less or equalto 1

6 is a sphere.

Proof. As both IG and J are symmetric with respect to the vertical line throughv = 1

2 , we will consider only volumes between 0 and 1/2. As the area of the sphereenclosing a volume 1/2 is larger than the one of the P minimal surface (which is! 2.34), we rule out the case that the entire profile IG comes from spheres. FromProposition 19, the maximum principle in Lemma 8 and item c) in Theorem 9 wededuce that there exists v1 < 1

2 such that IG(v) = Isph(v) for 0 $ v $ v1, and at thepoint q = (v1, IG(v1)) the spheres stop being isoperimetric and the higher topologybranch starts. The end point of this branch is (1

2 , IG(12)).

Figure 6. G-isoperimetric profile and supersolutions.

Consider a small volume v = 43$r

3 in the range where the symmetric profileIG agrees with the spherical one. Let *r be the solution of (23) starting at pr =(v, Isph(v)) = (4

3$r3, 4$r2) and tangent to the spherical profile at that point, see

Figure 6. According to (3) the graph of *r can be parameterized, for t # 0, as

vr(t) =4$

3r3 + 4$r2t + 4$rt2 ) 8$

3t3 , *r(vr(t)) = 4$r2 + 8$rt ) 8$t2.(24)

Since the spherical profile satisfies I2I !! + I(I !)2 ) 8$ = 0 and *r is a solution ofthe equation (23), we have from item iv) in Lemma 8 that *r(t) lies stricly belowIsph(v(t)) for t > 0.


For r small enough, the curve {(v, *r(v))|*r(v) # 0} lies below IG (in fact thiscurve converges to the origin when r goes to zero) and the highest point of *r ispmax(r) = (4$r3, 6$r2). Let

r0 = sup {r | 0 < r < r1 and *r(v) < IG(v) for all v $ 1

2such that *r(v) is defined},

where r1 is the radius of the last isoperimetric sphere (occurring at the branchingpoint q). So, there will be a first contact of *r0 with IG (outside the spherical part ofthe profile IG). This will occur either at q, or tangentially along the higher topologybranch, or at the end point (1

2 , IG(12)). We include the possibility of *r0 coinciding

with IG in the third case. The (strict) tangential contact cannot occur again becauseof the maximal principle, applied with the same constant # = )4, with IG being asupersolution of ()), and item b) in Theorem 9.

Contact at the branching point q is also impossible by the maximum principlein item iv) of Lemma 8 applied to the solution *r0 and the supersolution IG. Notethat, IG being trapped between two smooth curves whith are tangent at q, we havethat I !

G+ and *!r0coincide at this point. Moreover, item b) in Theorem 9 prevents

*r0 and IG coincide in an open interval).We are left with the third case, where *r0 hits the isoperimetric profile for the

half-volume (the two curves being possibly identical). Obviously the slope of *r0 atv = 1

2 is nonnegative, otherwise the curve would have points above the profile IG.Therefore, the maximum of *r0 is reached for a value of v not less than 1/2. Asthe maximum of *r is attained at v = 4$r3, there exists a special r2 $ r0 for which*!r2

(12) = 0, i. e. *r2 has its maximum at v = 1

2 . Furthermore *r2 lies under IG, likein Figure 6.

Using (24) we can find that

r2 =1

2$1/3& 0.341392,

so that the initial volume for *r2 is 16 and, therefore, *r2 is just the nonspherical

branch of J . This proves that IG # J . Moreover J(16) = $1/3 & 1.46459, while the

value at the maximum of J is J(12) = 6$r2

2 = 32$

1/3 & 2.19689.

In T = R3/Z3, we conjecture that the isoperimetric profile (for the nonsymmet-ric problem) coincides with the scp profile. From the theorem above we can nowconclude a property supporting this conjecture: a natural family of symmetric candi-dates can be excluded (namely, a one-parameter deformation of the classical Schwarzminimal surface by genus three symmetric surfaces with constant mean curvature,see [1]).

Corollary 21. The symmetric isoperimetric profile IG is not less than the spheres-cylinders-planes profile Iscp, and strictly greater when the volume is greater than 4!

81 ,i. e. in the cylindrical or planar range of Iscp. As consequence, isoperimetric surfaces


in the cubic torus (without prescribed extra symmetries) are not G-invariant, exceptfor the spheres.

Proof. Using the maximum principle in Lemma 8, it su&ces to check that IG liesabove Iscp at the transition points. At the transition point between spheres andcylinders, IG = Isph = Iscp because v = 4$/81 < 1

6 . At the other transition point, be-tween cylinders and planar tori, we have from Theorem 20 that IG(1/$) # J(1/$) >Iscp(1/$) = 2. The strict inequality can be easily computed numerically, but alsoproved by a slightly simpler argument, using the monotonicity of J : indeed J reachesheight 2 before Iscp. To see this fact, note that solving J(v(t)) = 2 yields

t =$2/3 )

'3$4/3 ) 4$

4$,

which corresponds to a volume

v(t) =1

2)

'3$1/3 ) 4

12'$

(2 + 3$1/3) & 0.311,

which is strictly less than 1/$ & 0.318 .

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Laurent [email protected] de Marne-la-ValleeChamps-sur-Marne, 77454 Marne-la-Vallee cedex 2France

Joaquin [email protected] de Geometrıa y TopologıaFacultad de Ciencias - Universidad de Granada18071 - Granada - Spain

Pascal [email protected] de Marne-la-ValleeChamps-sur-Marne, 77454 Marne-la-Vallee cedex 2France

Antonio [email protected] de Geometrıa y TopologıaFacultad de Ciencias - Universidad de Granada18071 - Granada - Spain

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