Double Bubbles in the Three-Torus Miguel Carrion´ Alvarez,´ Joseph Corneli, Genevieve Walsh, and Shabnam Beheshti
CONTENTS We present a conjecture, based on computational results, on 1. Introduction the area-minimizing method of enclosing and separating two 2. Preliminaries arbitrary volumes in the flat cubic three-torus, T3.Forcompa- 3. Some Examples and Construction Techniques rable small volumes, we prove that the standard double bubble 4. Simple Groups from R3 is area-minimizing. 5. Conclusions and Further Research Acknowledgments References 1. INTRODUCTION The classical isoperimetric problem seeks the least-area waytoencloseasingleregionofprescribedvolume. About 200 BC, Zenodorus argued that a circle is the least-perimeter enclosure of prescribed area in the plane (see [Heath 60]). In 1884, Schwarz [Schwarz 1884] proved by symmetrization that a sphere minimizes surface area for a given volume in R3. Isoperimetric problems arise naturally in many parts of modern mathematics; Ros provides a nice survey [Ros 01]. The double bubble problem is a two-volume general- ization of the classical isoperimetric problem. In 2000, Hutchings, Morgan, Ritor´e, and Ros proved that the standard double bubble was the unique least-area way to enclose and separate two volumes in R3 ([Hutchings et al. 00], [Hutchins et al. 02], [Morgan 00, Chapter 14]). In this paper, we present a conjecture, based on substan- tial numerical evidence, on the least-area way to enclose and separate two volumes in the three-torus.
Conjecture 1.1. (Central Conjecture 2.1.) The ten dif- ferent types of two-volume enclosures pictured in Figure 1 comprise the complete set of surface area-minimizing double bubbles for the flat cubic three-torus T3.
The numerical results summarized in Figure 2 indicate the volumes for which we conjecture each type of double bubble minimizes surface area. In Section 4, we show that for two small comparable 2000 AMS Subject Classification: Primary ; Secondary: volumes, the standard double bubble from R3 is optimal Keywords: keyword here in T3. Specifically, we prove the following theorem:
c AKPeters,Ltd. 1058-6458/2001° $ 0.50 per page Experimental Mathematics 12:1, page 79 80 Experimental Mathematics, Vol. 12 (2002), No. 1
Standard Double Bubble Delaunay Chain
Cylinder lens Cylinder Cross Double Cylinder
Slab Lens Center Bubble Slab Cylinder
Cylinder String Double Slab
FIGURE 1. Catalog of conjectured minimizers.
Theorem 1.2. Let M be a flat Riemannian manifold of of constant-mean-curvature surfaces meeting smoothly in dimension three or four such that M has compact quo- threes at 120◦ along smooth curves, which meet in fours tient by its isometry group. Fix λ (0, 1].Thenthere at a fixed angle of approximately 109◦ ([Taylor 76, The- ∈ is an ²>0,suchthatif0 We conclude the paper with some extensions to the primary conjecture that address noncubic tori and addi- 1.2 Recent Results tional volume constraints. In 2000, Hutchings, Morgan, Ritor´e, and Ros announced a proof that the standard double bubble, the familiar shape consisting of three spherical caps meeting one an- 1.1 Existence and Regularity other at 120-degree angles, provides the area-minimizing Using the language of geometric measure theory, bubble method of enclosing and separating two volumes in R3 clusters can be expressed as rectifiable currents, varifolds, ([Hutchins et al. 02], [Hutchings et al. 00]). The key or (M,²,δ)-minimal sets [Morgan 00]. In three dimen- features of the proof are a component bound developed sions, area-minimizing bubble clusters exist and consist by Hutchings [Hutchings 97, Theorem 4.2] and an insta- Carrion´ Alvarez´ et al: Double Bubbles in the Three-Torus 81 bility argument. Reichardt et al. [Reichardt et al. 03] 1.5 Plan of the Paper 4 extended this result to R . Section 2 reviews the methods that led to our Central In 2002, Corneli et al. [Corneli et al. 03] solved the Conjecture 2.1 and to Figures 1 and 2. Section 3 surveys double bubble problem for the flat two-torus, showing some sub-conjectures. Section 4 is the statement and that there are five types of minimizers (one of which only proof of the main theorem on small volumes. Section occurs on the hexagonal torus). The proof relies on reg- 5 shifts from the cubic torus to other tori and discusses ularity, a variational component bound due to Wichira- other conjectures and candidates, including a “Hexagonal mala [Morgan and Wichiramala 02], and combinatorial Honeycomb.” and geometric classification. 2. THE CONJECTURE 1.3 Double Bubbles in T3 2.1 Generating Candidates Even the single bubble for the three-torus is not yet com- Participants in the CMI/MSRI Summer School proposed pletely understood. There are, however, partial results. many possibilities for double bubbles in the three-torus The smallest enclosure of half of the volume of the torus T3in brainstorming sessions. was shown by Barthe and Maurey [Barthe and Maurey 00, Section 3] to be given by two parallel two-tori. Mor- Standard Double Bubble. The least-area double bubble gan and Johnson [Morgan and Johnson 00, Theorem 4.4] in R3. showed that the least-area enclosure of a small volume is a sphere. Spheres, tubes around geodesics, and pairs Delaunay Chain. Two stacked “beads” or “drums” of parallel two-tori are shown to be the only types of wrapping around one of the periods of the torus. area-minimizing enclosures for most three-tori by work The lateral surfaces are Delaunay surfaces (constant- of Ritor´e and Ros ([Ritor´e 97, Theorem 4.2], [Ritor´eand mean-curvature surfaces of rotation). Ros 96]). Cylinder Lens. A cylinder wrapping around a period of the torus, with a small bubble attached. 1.4 Small Comparable Volumes Double Cylinder. Two cylinders wrapping around a pe- We prove Theorem 4.1 by showing that every sequence of riod of the torus. Transverse sections by an orthogo- area-minimizing double bubbles with decreasing volume nal plane (i.e., a two-torus) are the standard double and fixed volume ratio contains standard double bubbles. bubble in two dimensions. This is the first of a fam- The main difficulty lies in bounding the curvature. Once ily of solutions we could describe as “cylinders over this is accomplished, it is possible to show that the bubble minimizing double bubbles in the two-torus.” lies inside a small ball that lifts to R3, where by [Hutchins et al. 02], a minimizer is known to be standard. Cylinder String. A cylinder over a “Symmetric Chain” 2 The main idea of the argument is as follows: From a in T . given sequence of double bubbles with shrinking volumes Slab Cylinder. Two parallel flat two-tori with a cylindri- and fixed volume ratio, we generate a new sequence by cal bubble attached to one of them. This is a cylin- rescaling the ambient manifold at each stage so that one derovera“BandLens”inT2. of the volumes is always equal to one. We can then ap- ply compactness arguments and area estimates to show Double Slab. Three parallel flat two-tori (cylinder over that certain subsequences of sequences obtained by rigid a “Double Band” in T2). motions of the ambient manifold have nontrivial limits. Slab Lens. Two parallel flat two-tori, one of which has a These limits are used to obtain a curvature bound on a small lens-shaped bubble stuck in it. subsequence. With such a bound, we apply a monotonic- ity result to conclude that for small volumes, the double Center Bubble. Two close-to-parallel two-tori with a bubble is contained in a small ball. We conclude that it close-to-cylindrical bubble between them. The must be the same as the minimizer in R3 or R4, i.e., it planes buckle slightly (as can be shown by elemen- must be the standard double bubble by [Hutchins et al. tary stability considerations) and the bubble stuck 02] or [Reichardt et al. 03]. The result extends to any between them is not quite round (it bulges slightly flat three- or four-manifold with compact quotient by the in the direction of each of the four corners of the isometry group. fundamental domain pictured in Figure 1). 82 Experimental Mathematics, Vol. 12 (2002), No. 1 SDB = Standard Double Bubble DC = Delaunay Chain CL = Cylinder Lens CC = Cylinder Cross 2C = Double Cylinder SL = Slab Lens CB = Center Bubble CS = Cylinder String SC = Slab Cylinder 2S = Double Slab FIGURE 2. Phase portrait: volumes and corresponding double bubble. In the center both regions and the complement have one third of the total volume; along the edges, one volume is small; in the corners, two volumes are small. Transverse Cylinders. Two cylinders wrapping around Cylinder Cross. A cylinder wrapping around one period two different periods of the three-torus that touch of the torus with an attached bubble that wraps each other to reduce surface area. around one of the perpendicular directions. (Ob- tained from the Cylinder Lens when the small region Hydrant Lens. Known as “Scary Gary” after its inven- grows.) tor, Gary Lawlor; this is essentially the Schwarz P surface with a small bubble attached. Center Cylinder. Two parallel flat two-tori with a cylin- der going through both of them and wrapping Double Hydrant. Another fanciful proposal based on the around a period of the three-torus. (Obtained from Schwarz P surface. the Center Bubble when the small region grows or from the Transverse Cylinders as one of the cylinders Inner Tube. A cylinder that wraps around one of the grows.) periods of the three-torus with a toroidal bubble wrapped around it. 2.2 Producing the Phase Diagram Brakke’s Surface Evolver [Brakke xx] was used to closely When preparing for the calculations that resulted in approximate the minimal area that a double bubble of thephasediagramofFigure2,weanticipatedthepossi- each type needs to enclose specified volumes. Our initial bility that a minimizing bubble topology might have been simulationsweredonewithvolumeincrementsof0.05 missed. In fact, two additional candidate topologies were (from a total volume of 1) for the initial set of brain- found by systematic application of this heuristic: stormed candidates. The results of these calculations When the two volumes are very small, the stan- were sufficient to convince us that the Hydrant Lens and dard double bubble should be optimal. As one theDoubleHydrantwereinefficient and that the Inner or both of the two volumes grow, a bubble that Tubewasinfactunstable(whichiswhywecouldnot collides with itself should open up to reduce draw a picture of it with Surface Evolver for inclusion in perimeter, whereas if two different bubbles col- this paper). A rough version of Figure 2 was then ob- lide, they should stick. tained using a volume increment of 0.01. From the data obtained in these computations, we found that each of This procedure incorporates the assumption that all the double bubbles pictured in Figure 1 was the least-area three regions associated with a minimizer will be con- competitor for some volume triple v1 : v2 : v3. The phase nected. The new candidates we found were as follows: diagram appearing in Figure 2 is the result of refining Carrion´ Alvarez´ et al: Double Bubbles in the Three-Torus 83 Transverse Cylinders Double Hydrant Center Cylinder Hydrant Lens FIGURE 3.Inefficient double bubbles. our second series of computations along the boundaries // five cycles of refining the mesh and between the regions using a volume increment of 0.005. // decreasing area {{u; g; u} 10; {u; r; u}} 5; // decrease area ten more times at the end Conjecture 2.1. (Central Conjecture.) The ten dou- {u; g; u} 10; ble bubbles pictured in Figure 1 represent each type of surface area-minimizing two-volume enclosure in a flat, These commands produced pictorial output that per- cubicthree-torus,andthesetypesareminimizingforthe fectly matches our idea of what each conjectured mini- volumes illustrated in Figure 2. mizer should look like. 2.3 Comments One might guess that minimizers would be formed from 3. SUBCONJECTURES topological spheres and tori that go around at least one We now present a list of natural subconjectures about 3 period of T . This was borne out in our computations. It bubbles in the flat cubic three-torus T3 that are sug- is interesting to note that although the Transverse Cylin- gested either by the phase diagram or by examination of ders pictured in Figure 3 match this description, they the pictures made by Surface Evolver. were never seen to be minimizing. One immediate observation is that the edges of our A challenging problem related to the problem of area phase diagram appear to characterize single bubbles minimization is that of finding all of the stable dou- in T3. ble bubbles in T3; there are probably quite a few that we have not looked at. Note that a given type from Conjecture 3.1. (Ritor´eandRos[Ritor´e97], [Ritor´eand Figure 1 might be stable for a much wider range of vol- Ros 96], [Ros 01].) The optima for the isoperimetric umes than those for which it actually minimizes surface problem in T3 are the sphere, cylinder, and slab. area. It is worth observing that all of the conjectured mini- The following two conjectures are perhaps the most 2 mizers for the double bubble problem on T are echoed intuitive things we would expect to be true about bubbles here in at least two ways: several conjectured minimiz- in T3: ers are T2 minimizers S1, and others are more di- × rect analogues, for example, the Delaunay and Symmet- Conjecture 3.2. An area-minimizing double bubble in T3 ric Chains. See Corneli et al. [Corneli et al. 03] for more has connected regions and complement. on the T2 minimizers. A few words about the accuracy of the simulations Conjecture 3.3. There is an ²>0 such that if v1,v2 are are in order. The commands used in our final stage of both less than ², the standard double bubble of volumes v1 numerics were as follows: 3 and v2 is optimal in T . 84 Experimental Mathematics, Vol. 12 (2002), No. 1 In Theorem 4.1, we prove that there is such an ² for is a pair of parallel two-tori that divide both volumes in every fixed ratio of volumes. half. If it were known for some reason that the number of components of a minimizer was relatively small, then we Proof: This is a generalization of the standard argument think the next two conjectures might admit fairly easy for the ham sandwich theorem in Euclidean space. We proofs: will regard the three-torus as the result of identifying the faces of a rectangular cube [ n, n] [ m, m] [0,p]in Conjecture 3.4. For one very small volume and two mod- − × − × R3. We may assume that the solid torus is a vertical erate volumes, the Slab Lens is optimal. cylinder with base centered at the origin in the x y − plane. Now we rotate this solid cylinder, and index the The first phase transition as small equal Conjecture 3.5. rotation by α [0, 2π]. Then for each α [0, π], rotate or close to equal volumes grow is from the standard double ∈ ∈ the cylinder (along with the surface that lies inside) by bubble to a chain of two bubbles bounded by Delaunay α in the x y plane and consider the family of pairs of surfaces. − planes y = c, y = c + m ,wherec [ m, 0] For each { } ∈ − Delaunay surfaces are constant-mean-curvature sur- α [0, π],thereisatleastonesuchpairofplanesthat ∈ faces of rotation, and as such contain an S1 in their cuts the volume V1 in half, in the sense that half of V1 is symmetry group. From the Surface Evolver pictures, it in the inside of the planes, and half of V1 is on the outside appears that the conjectured surface area minimizers al- of the two planes. There may be an interval of such pairs ways have the maximal symmetry, given the constraints, for a given α. However, some one parameter family of a fact that leads to the following natural conjecture. these planes may be chosen which varies continuously as a function of α, and coincides at 0 and π.Sincebythe Conjecture 3.6. All isotopies that are not isometries de- time α reaches π the two portions of V2 have switched crease a minimizer’s symmetry group. sides, there must be some pair of planes that divides V2 We will conclude the section with a proposition that in half. The restriction of these planes to the domain [ n, n] [ m, m] [0,p] glue up to a pair of two-tori in establishes a very limited symmetry property for all dou- − × − × ble bubbles in the two torus (including nonminimizing the three-torus that divides both regions in half. bubbles). Specifically, we prove that for any double bub- ble, there is a pair of parallel two-tori that cut both re- gions in half. Hutchings [Hutchings 97, Theorem 2.6] was able to show that in Rn a minimizing double bubble 4. SMALL COMPARABLE VOLUMES has two perpendicular planes that divide both regions in half. He used this result in the proof that the function Conjecture 3.3 asserts that two small volumes in T3 are that gives the least-area to enclose two given volumes in best enclosed by a standard double bubble. In this sec- Rn is concave. We conjecture that a similar concavity tion, we will prove Theorem 4.1, which says that for any result also holds for the three-torus: fixed volume ratio, the standard double becomes optimal when the volumes are sufficiently small. This result holds Conjecture 3.7. The least area to enclose and separate for a relatively broad class of three- and four-dimensional two given volumes in the three-torus is a concave function manifolds (and see Remark 4.4). of the volumes. The standard double bubble consists of three spherical caps meeting at 120 degrees (see Figure 2). If the volumes IfonecouldproveConjecture3.7,onewouldthenbe are equal, the middle surface is planar. The standard able to apply other ideas in Hutchings’ paper [Hutchings double bubble is known to be minimizing for all volume 97,Section4]toobtainafunctionalboundonthenumber pairs in R3 [Hutchins et al. 02] and R4 [Reichardt et al. of components of a minimizing bubble. 03]. Proposition 3.8. (Deluxe Ham Sandwich Theorem.)∗ If adoublebubbleliesinsideasolidtwo-torusinsidearec- Theorem 4.1. Let M be a flat Riemannian manifold of tangular three-torus, D2 S1 S1 S1 S1, then there dimension three or four such that M has compact quo- × ⊂ × × tient by its isometry group. Fix λ (0, 1].Thenthere ∈ is an ²>0,suchthatif0 Remark 4.2. Our assumption on the isometry group The second step in the proof is to use these limits to guarantees solutions to the double bubble problem for obtain a weak bound on the curvature of a subsequence all volume pairs (the proof is the same as in Morgan of Sv. We need this in order to apply the monotonicity [Morgan 00, Section 13.7]). Not every flat 3-manifold theorem for mass ratio [Allard 72, Section 5.1(1)], which has compact quotient by its isometry group. Euclidean says that a small ball around any point on a surface with 3-space modulo a glide reflection or a glide rotation is an weakly bounded mean curvature contains some substan- example. tial amount of area. The third step is to show that all of the surface area It is helpful to have the overall strategy of the proof of each element of our subsequence is contained in some in mind before we go on. Our goal is to prove that a ball in Mv that has fixed radius for all v. Eventually, as minimizing double bubble of volumes v, λv lies inside a v shrinks and Mv grows, this ball will have to be trivial trivial ball in M when v is small enough. The results for in Mv. Euclidean space then prove our claim. We then use the result that the optimal double bub- For the proof, we will regard a double bubble as a pair ble in R3 is standard to show that our subsequence has a of three- or four-dimensional rectifiable currents, R1 and tail comprised of standard double bubbles. The desired R2, each of multiplicity one, of volumes V1 = M(R1) conclusion then follows from the fact that Sv is simply and V2 = M(R2). The total area of such a double bub- a scaled version of the original double bubble containing 1 ble is 2 (M(∂R1)+M(∂R2)+M(∂(R1 + R2))). Here M volumes v and λv. Thus there is no sequence of shrink- denotes the mass of the current, which can be thought ing fixed-ratio area-minimizing double bubbles in M that of as the Hausdorff measure of the associated rectifiable does not contain standard double bubbles. set (counting multiplicities). For a review of the perti- We focus on the case of dimension three; the proof for nent definitions, see the notes from Morgan’s course at dimension four is essentially identical. the CMI/MSRI Summer School [Morgan and Ritor´e01] The following lemma is needed in the first step of the or the texts by Morgan [Morgan 00] or Federer [Federer proof. 69]. By the Nash and subsequent isometric embedding theorems, we may assume M is a submanifold of some Lemma 4.3. There is a γ > 0 such that if R is a region fixed RN . in an open Euclidean 3-cube K and vol(R) vol(K)/2, ≤ We will consider a sequence of area-minimizing double then bubbles in M enclosing the volumes v and λv,asv 0. 2/3 → area(∂R) γ(vol(R)) . If the theorem were false, there would be some such se- ≥ quence with no standard double bubbles in it. We prove that every sequence of fixed volume ratio double bubbles Proof: Let γ0 be such an isoperimetric constant for a 2/3 with shrinking volumes contains standard double bub- cubic three-torus, so that area(∂P ) γ0(vol(P )) for 3 ≥ bles. all regions P T .Suchaγ0 exists by the isoperimet- N ⊆ For each v, Mv will denote sv(M)inR ,wheresv ric inequality for compact manifolds [Morgan 00, Section is the scaling map that takes regions with volume v to 12.3]. Make the necessary reflections and identifications similar regions with volume 1. In particular, sv maps of the cube K toobtainatoruscontainingaregionR0 our area-minimizing double bubble enclosing volumes v with eight times the volume and eight times the surface and λv to a double bubble which we call Sv that encloses area of R. The claim follows, with γ = γ0/2. volumes 1 and λ, and is of course area-minimizing for these volumes. Proof of Theorem 4.1: The firststepistoshowthatonceMv has been suit- Step 1. The sequence Sv Mv,whereeachMv has ⊂ ably translated and rotated, the sequence Sv has a subse- been suitably translated and rotated, has a subsequence quence that converges as v 0withV1 = 0 in the limit. that converges with V1 =0in the limit. → 6 6 Our argument also shows that there is a differently mod- We first show the existence of a covering Kv of Mv ified subsequence that converges to a limit with V2 =0, with bounded multiplicity, consisting of 3-cubes con- 6 but does not show that there is a subsequence where both tained in Mv, each of side-length L, for any L>0. volumes are nonzero in the limit. This is because while Lemma 4.3 will give us a positive lower bound on the we are exerting ourselves trapping the first volume in a volume of the part of R1 that is inside one of these cubes ball, the second one may wander off to infinity. for each Sv. We will then apply a standard compactness 86 Experimental Mathematics, Vol. 12 (2002), No. 1 theorem to show that a subsequence of the sequence of Since D is contained in the limit of the Mv,namely, 3 Sv converges. the copy of R chosen above, and each Sv is minimizing 1 Take a maximal packing of Mv by balls of radius 4 L. for its volumes, a standard argument shows that the limit 1 Enlargements of radius 2 L cover Mv. Circumscribed 3- D is the area-minimizing way to enclose and separate 3 3 cubes in Mv of edge-length L provide the desired cover- the given volumes vol(R1) δ and vol(R2)inR (see ≥ ing Kv. To see that the multiplicity of this covering is [Morgan 00, 13.7]). In the limit, vol(R2) could be zero, bounded, consider a point p Mv. The ball centered at in which case D is a round sphere. If both volumes are ∈ p with radius 2L contains all the cubes that might cover nonzero, D is the standard double bubble ([Hutchins et 1 it, and the number of balls of radius 4 L that can pack al. 02] and [Reichardt et al. 03], or see [Morgan 00, into this ball is bounded, implying that the multiplicity Chapter 14]). of Kv is also bounded by some m>0. Step 2. By taking a subsequence if necessary, we may Now let Kv beacoveringasabove,withL =2.By assume there is a weak bound on the curvature of the Sv. Lemma 4.3, there is an isoperimetric constant γ such that To show that the mean curvature is weakly bounded is to show that there is a C such that for any Sv,andany 2/3 area(∂(R1 Ki)) γ(vol(R1 Ki)) , ∩ ≥ ∩ direction in the two-dimensional space of possible volume dA changes, dV C for smooth variations. Thus it suffices and therefore, since maxk vol(R1 Kk) vol(R1 Ki) | | ≤ ∩ ≥ ∩ to produce smooth variations such that changes in the for any i, volume of Sv and in the area of ∂Sv are controlled, that vol(R1 Ki) is, that the magnitude of the change in volume is bounded area(∂(R1 Ki)) γ ∩ 1/3 . (4—1) ∩ ≥ (maxk vol(R1 Kk)) from below and the magnitude of the change in area is ∩ bounded from above. Note that the total area of the surface is greater than We first show that the magnitude of the change in 1/m times the sum of the areas in each cube, and the volume is bounded below. Take a smooth variation vector total volume enclosed is less than the sum of the volumes, field F in Rn such that for D, so summing Equation 4—1 over all the cubes Ki in the covering K yields v dV1/dt = (F n) dA = c =0 ∂R1 · 6 V Z area(S ) area(∂R ) mγ 1 v 1 1/3 and ≥ ≥ (maxk vol(R1 Kk)) ∩ dV2/dt =0. and (Note that we need the firstvolumetobenonzero,orits 1/3 V1 variation could be zero.) For v small enough, the subse- (max(vol(R1 Kk))) mγ δ, k ∩ ≥ area(Sv) ≥ quence of Sv associated with the limit D has the property that dV /dt is approximately equal to the constant c and for some δ > 0, because V = 1 and area(S ) is bounded 1 1 v dV /dt is approximately equal to 0. (there is a bounded way of enclosing the volumes, and 2 By the argument in the first step, we can translate and Sv has the same amount of area or less). rotate each Mv so that, after taking a subsequence again Translate each Mv so that a cube Kk that maximizes 3 N if necessary, Sv converges to a minimizer D0 in R where vol(R1 Kk)iscenteredattheoriginofR ,androtate ∩ the second volume is nontrivial. This time we take a so that the tangent space of each M at the origin is equal v smooth variation vector field F , such that the images of to a fixed R3 in RN . The limit of the M will be equal 0 v S under the appropriate rigid motions have the property to this R3. Since a cube with edge-length L centered at v that dV /dt is approximately equal to 0 and dV /dt is the origin fits inside a ball of radius 2L centered at the 1 2 approximately equal to some constant c0 =0,forv small origin, we have 6 enough. 3 Note that the change in volume is independent of rigid vol(R1 B(0, 2L)) δ ∩ ≥ motions; in particular, we can translate both F and F 0 for every Sv. By the compactness theorem for locally back along the rigid motions to act on Sv.Thisproves integral currents ([Morgan 00, pp. 64, 88], [Simon 84, that for this sequence, the magnitude of the change in Section 27.3, 31.2, 31.3]), we know that a subsequence of volume is bounded below. the Sv has a limit, which we will call D, with the property Now we need to show that the change in area is 3 that vol(R1) δ . This completes the first step. bounded above. This follows from the fact that every ≥ Carrion´ Alvarez´ et al: Double Bubbles in the Three-Torus 87 rectifiable set can be thought of as a varifold [Morgan double bubbles. We conclude that the minimizing dou- 00, Section 11.2]. By compactness for varifolds [Allard ble bubble with volume ratio λ is standard when v is 72, Section 6], the Sv, situated so that the first volume small enough. does not disappear, converge as varifolds to some varifold, J.Thefirst variations of the varifolds also converge, i.e., Remark 4.4. Given n and m, essentially the same ar- δSv δJ; see [Allard 72]. The first variation of a varifold gument shows that for any smooth n-dimensional Rie- → is a function representing the change in area. Therefore, mannian manifold with compact quotient by the isome- far enough out in the sequence the change in area of the try group, given 0 < λ 1, there are C, ²>0, such ≤ Sv under F is bounded close to the change in area of J that for any 0 Manuel Ritor´e and John Sullivan for several helpful conversa- tions. Other participants who made notable contributions in- clude Baris Coskunuzer, Brian Dean, Dave Futer, Tom Flem- ing, Jim Hoffman, Jon Hofmann, Paul Holt, Matt Kudzin, Jesse Ratzkin, Eric Schoenfeld, and Jean Steiner. Examining the results on T2 proved to be quite helpful to us in our work on the T3 project. We thank our colleagues from the SMALL Geometry Group for sharing their insights. Yair Kagan of New College of Florida created the picture of the “Hexagonal Honeycomb” that appears in Figure 4. We thank him for this contribution. FIGURE 4. Hexagonal honeycomb We thank Ken Brakke for sharing his wonderful program, Surface Evolver , with the world and for answering questions about its use. [Adams et al. 2003], by Lebesgue’s Covering Theorem The second author participated in the 2001 SMALL un- [Hurewicz and Wallman 41, Theorem IV2], such regions dergraduate summer research Geometry Group, supported by Williams College and by the National Science Foundation mustsometimesmeetinfoursormore. throughanREUgranttoWilliamsandanindividualgrant This suggests that it is likely that there are no other to Morgan. Genevieve Walsh was supported in part by NSF special minimizers for the double bubble problem. VIGRE Grant No. DMS-0135345. Conjecture 5.3. The double bubbles of Figure 1 together REFERENCES with the Hexagonal Honeycomb of Figure 4 comprise the [Adams et al. 2003] Colin C. Adams, Frank Morgan, and complete set of area-minimizing double bubbles for all John M. Sullivan. “When Soap Bubbles Collide.” three-tori. Preprint, 2003. [Allard 72] William K. Allard. “On the First Variation of a In light of the fact that the triple bubble problem in Varifold.” Ann. of Math.(2) 95 (1972), 418—446. the torus seems likely to produce many interesting can- [Barthe and Maurey 00] Franck Barthe and Bernard Mau- didates, we conclude with one final conjecture. rey. “Some Remarks on Isoperimetry of Gaussian Type.” Ann. Inst. H. Poincar´eProbab.Statist.36 (2000), 419— Conjecture 5.4. For the triple bubble problem in a flat 434. cubic three-torus in the case where one of the volumes is [Brakke xx] Kenneth A. Brakke. Surface Evolver. Avail- small, the minimizers will look like the double bubbles of able from World Wide Web (http://www.susqu. edu/facstaff/b/brakke/evolver/evolver.html), xxxx. Figure 2 with a small ball attached. The phase diagram will look just like our Figure 3. [Corneli et al. 03] Joseph Corneli, Paul Holt, George Lee, Nicholas Leger, Eric Schoenfeld, and Benjamin Stein- hurst. “The Double Bubble Problem on the Flat Two- Torus.” Preprint, 2003. ACKNOWLEDGMENTS [Federer 69] Herbert Federer. Geometric Measure Theory. Berlin: Springer-Verlag, 1969. This paper was prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global [Heath 60] Thomas Heath. A History of Greek Mathematics, Theory of Minimal Surfaces, held at the Mathematical Sci- Volume II. Oxford, UK: Oxford University Press, 1960. ences Research Institute in Berkeley, California, Summer [Hurewicz and Wallman 41] Witold Hurewicz and Henry 2001, and has its origins in a problem given there by Frank Wallman. Dimension Theory.Princeton,NJ:Princeton Morgan in his course, “Geometric Measure Theory and the University Press, 1941. Proof of the Double Bubble Conjecture.” We are very grate- [Hutchings 97] Michael Hutchings. “The Structure of Area- ful to Frank Morgan for taking the time to guide us through Minimizing Double Bubbles.” J. Geom. Anal. 7 (1997), this project with many helpful discussions and constructive 285—304. comments. ThankstoCMIandMSRIformakingthesummerschool [Hutchings et al. 00] Michael Hutchings, Frank Morgan, possible, and for providing research and travel support to the Manuel Ritor´e, and Antonio Ros. “Proof of the Dou- authors. We extend our thanks to Joel Hass, David Hoff- ble Bubble Conjecture.” Electron. Res. Announc. Amer. man, Arthur Jaffe, Antonio Ros, Harold Rosenberg, Richard Math. Soc. 6 (2000), 45—49 (electronic). Schoen, and Michael Wolf, who organized the summer school. [Hutchins et al. 02] Michael Hutchings, Frank Morgan, We thank faculty participants Michael Dorff,DeniseHalver- Manuel Ritor´e, and Antonio Ros. “Proof of the Double son, and Gary Lawlor for help with the initial simulations, and Bubble Conjecture.” Ann. Math. 155 (2002), 459—489. Carrion´ Alvarez´ et al: Double Bubbles in the Three-Torus 89 [Kusner and Sullivan 96] Robert B. Kusner and John M. Sul- [Reichardt et al. 03] Ben W. Reichardt, Cory Heilmann, livan. “Comparing the Weaire-Phelan Equal-Volume Yuan Y. Lai, and Anita Spielmann. “Proof of the Dou- Foam to Kelvin’s Foam.” Forma 11 (1996), 233—242. ble Bubble Conjecture in R4 and Certain Higher Dimen- [Morgan 00] Frank Morgan. Geometric Measure Theory: A sional Cases.” Pacific J. Math. 208 (2003), 347—366. Beginner’s Guide, Third edition. San Diego, CA: Acad- [Ritor´e97]ManuelRitor´e. “Applications of Compactness Re- emic Press, Inc., 2000. sults for Harmonic Maps to Stable Constant Mean Cur- [Morgan 01] Frank Morgan. “Small Perimeter-Minimizing vature Surfaces.” Math. Z. 226 (1997), 465—481. Double Bubbles in Compact Surfaces are Stan- [Ritor´eandRos96]ManuelRitor´e and Antonio Ros. “The dard.” In Electronic Proceedings of the 78th Annual Spaces of Index One Minimal Surfaces and Constant meeting of the Lousiana/Mississippi Section of the Mean Curvature Surfaces Embedded in Flat Three Man- MAA. Univ. of Miss., March 23-24, 2001. Avail- ifolds.” Trans. Amer. Math. Soc. 348 (1996), 391—410. able from World Wide Web (http://www.mc.edu/ [Ros 01] Antonio Ros. “The Isoperimetric Problem.” Lec- campus/users/travis/maa/proceedings/spring2001/), ture notes on a series of talks given at the Clay Mathe- 2001. matics Institute Summer School on the Global Theory [Morgan and Johnson 00] Frank Morgan and David L. John- of Minimal Surfaces June 25, 2001 to July 27, 2001, son. “Some Sharp Isoperimetric Theorems for Rie- at the Mathematical Sciences Research Institute, Berke- mannian Manifolds.” Indiana U. Math J. 49 (2000), ley, California, 2001. Available from World Wide Web 1017—1041. (http://www.ugr.es/˜aros/isoper.pdf), 2001. [Morgan and Ritor´e 01] Frank Morgan and Manuel Ritor´e. [Schwarz 1884] Hermann A. Schwarz. “Beweis des Satzes, “Geometric Measure Theory and the Proof of the Double dass die Kugel kleinere Oberfl¨ache besitzt als jeder Bubble Conjecture.” Lecture notes by Ritor´eonMor- andere K¨orper gleichen Volumens.” Nachrichten gan’s lecture series at the Clay Mathematics Institute K¨oniglichen Gesellschaft Wissenschaften G¨ottingen Summer School on the Global Theory of Minimal Sur- (1884), 1—13. faces, at the Mathematical Sciences Research Institute, [Simon 84] Leon Simon. “Lectures on Geometric Measure Berkeley, California 2001, Available from World Wide Theory.” In Proc. Centre Math. Anal. Australian Nat. Web (http://www.ugr.es/˜ritore/preprints/course.pdf), U. Vol. 3. Canberra, Australia: Centre for Mathematical 2001. Analysis, Australian National University, 1984. [Morgan and Wichiramala 02] Frank Morgan and Wacharin [Taylor 76] Jean E. Taylor. “The Structure of Singularities Wichiramala. “The Standard Double Bubble is the 2 in Soap-Bubble-Like and Soap-Film-Like Minimal Sur- Unique Stable Double Bubble in R .” Proc. AMS 130 faces.” Ann. of Math. (2) 103:3 (1976), 489—539. (2002), 2745—2751. Miguel Carri´on Alvarez,´ University of California, Riverside ([email protected]) Joseph Corneli, University of Texas, Austin, Department of Mathematics, RLM 8.100, Austin, TX 78712 ([email protected]) Genevieve Walsh, University of California, Davis ([email protected]) Shabnam Beheshti, University of Massachusetts, Amherst ([email protected]) Received March 25, 2002; accepted in revised form April 10, 2003.