Double Bubbles in the Three-Torus Miguel Carrion´ Alvarez,´ Joseph Corneli, Genevieve Walsh, and Shabnam Beheshti
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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 <v<², an area-minimizing orems II.4, IV.5, IV.8], or [Morgan 00, Section 13.9]). double bubble in M of volumes v, λv is standard. 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.