c Copyright by Wacharin Wichiramala, 2002

Transcription

1 c Copyright by Wacharin Wichiramala, 2002

2 THE PLANAR TRIPLE BUBBLE PROBLEM BY WACHARIN WICHIRAMALA B.Sc., Chulalongkorn University, 1994 M.S., University of Illinois at Urbana-Champaign, 1997 THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2002 Urbana, Illinois

3 Abstract We prove the planar triple bubble conjecture that the standard triple bubble is the unique leastperimeter way to enclose and separate three regions of given areas. We also prove a bound on the number of convex components and a bound on the number of all components in any minimizing planar bubble. iii

4 To my mother. iv

5 Acknowledgments The author is very grateful to his advisor Prof. John M. Sullivan for his guidance and fruitful discussions during the preparation of this thesis. Thanks are also due Richard Vaughn, Michael Hutchings, Angel Montesinos Amilibia and Manuel Ritoré for their helpful communication. Contribution from Prof. Stephanie B. Alexander, Prof. I. David Berg, Prof. Igor G. Nikolaev and Prof. Christopher P. French as members of the author s final examination committee is much appreciated. The author would like to express his gratitude to Dr. Porama Saengcharoenrat and Prof. Aimo Hinkkanen for their mentoring and encouragement. In addition, the author would like to thank the DPST (Development and Promotion of Science and Technology Talent Project) scholarship of Thailand, and NASA grant NAG for partial support in summer Lastly, the author would like to mention the great support from his wife Yupin Patarapongsant and their children Ken and Pam. v

6 Table of Contents Chapter 1 Introduction Chapter 2 Preliminary results Existence and regularity of planar soap bubbles Results on minimizing bubbles Geometry of standard bubbles Weak approach to the planar triple bubble conjecture Chapter 3 Basic results Conditions on bubbles The weak approach Using the weak approach Chapter 4 Bounds on the number of components Examples of a nonminimizing bubble Variations of bubbles Second variation formula Variations of convex components Variations on long edges Bounds of number of convex components and long edges Bounds for the total number of components Chapter 5 Geometry of planar soap bubbles Geometry of planar bubbles Reduction and decoration on bubbles Geometry of triple bubbles Chapter 6 Planar triple bubble conjecture Categorization of small triple bubbles List of potential weakly minimizing triple bubbles Potential weakly minimizing totally reducible triple bubbles Potential weakly minimizing decorations of irreducible bubble Potential weakly minimizing decorations of irreducible bubble 4A and 10C Potential weakly minimizing decorations of irreducible bubble 4B Potential weakly minimizing decorations of irreducible bubble Potential weakly minimizing decorations of irreducible bubble Potential weakly minimizing decorations of irreducible bubble 10A vi

7 6.2.8 Potential weakly minimizing decorations of irreducible bubble 10B Potential weakly minimizing decorations of irreducible bubble 10D Potential weakly minimizing decorations of irreducible bubble 11A and 11B Potential weakly minimizing decorations of irreducible bubble 12A Potential weakly minimizing decorations of irreducible bubble 12B Potential weakly minimizing decorations of irreducible bubble Potential weakly minimizing decorations of irreducible bubble Conclusion Eliminating nonstandard potential bubbles Variation formulas Crucial configurations Eliminating nonstandard bubbles The planar triple bubble theorem Chapter 7 Future study Standard triple bubbles Geometry of standard triple bubbles Efficiency of standard triple bubbles On the planar soap bubble conjecture Related topics and difficulty Using simply overlapping bubbles to prove the triple bubble theorem Conjectural isoperimetric inequality More on bubble variations Stability of some certain bubbles Variations on hexagonal components Shape of minimizing bubbles Miscellaneous improvements References Vita vii

8 Chapter 1 Introduction Humans have been fascinated by soap bubbles in many ways since ancient time. One aspect is that they are believed to minimize surface area while enclosing fixed volumes of air. The planar soap bubble problem is the mathematically analogous problem in two dimensions: the search for the least-perimeter way to enclose and separate regions of given areas on the plane. It has been known since the time of the ancient Greeks that a circle is the best way to enclose a single given area, but this was not proved rigorously until much later in the late nineteenth century. Intuitively, it seems clear that it should be best to keep each region in a single connected component. But mathematically, this is the main difficulty: there is no a priori way to show that it is not better to split one region into several smaller components. For any dimension greater than two, the soap bubble problem was proven to have a solution by Almgren [Almgren] in He also gave some basic regularities of such solutions. In the same year, Taylor [Taylor] added more regularity for the case of three dimensions that minimizing soap bubbles have structure we are familiar with. In 1985, Bleicher [Bleicher1] proved a great regularity properties of solutions to the problem in 2 dimensions without giving a rigorous proof of their existence. In 1992, Morgan [Morgan] proved the existence of solutions to the planar problem with a new regularity result. A year later, in 1993, a group of undergraduate students, Foisy, Alfaro, Brock, Hodges and Zimba [FABHZ], solved the planar problem of 2 areas. They found a new approach to eliminate the possibility of having empty chambers. A year later, in 1994, another group of undergraduate students, Cox, Harrison, Hutchings, Kim, Light, Mauer and Tilton [CHHKLMT], started the case of 3 areas as they proved that, among enclosures of connected regions (including the exterior region), the standard triple bubble is uniquely shortest. The most impressive result from this paper is the understanding between pressures of regions and signed curvatures of edges. In 1996, Bleicher proved a powerful result in [Bleicher3] that, in a minimizing bubble, any 2 components may meet at most once. This reduces many combinatorial possibilities for candidate bubbles as it gives the sense of uniqueness of the edge between 2 components. In 1998, Vaughn finished his Ph.D. dissertation focusing on the planar triple bubble conjecture. He explicitly showed why the new approach is valid for 3 areas and cleverly proved that any minimizing triple bubble with equal pressures and without empty chamber is standard. From these results, in order to prove the conjecture, we just need to show that any nonstandard triple bubble with unequal pressures and without empty chambers is not minimizing in the weak sense. In chapter 2, we conclude all preliminary results together with important definitions that will 1

9 be used throughout this dissertation. The last section shows a powerful approach to the planar triple bubble conjecture, called the weak approach. It helps eliminate the possibility of having empty chambers by allowing a bubble to enclose greater areas than prescribed. In chapter 3, we list some basic results. The first section shows some results that are simple but colorful in some sense. The second section analyzes the weak approach. We show that the approach is valid for the planar problem of up to 6 areas and for the higher dimensional problem of a small number of volumes. The last section obtains some results from the weak approach such as a bound on the number of components. However in the next chapter we will have much better bounds. In chapter 4, we develop new component bounds for minimizing planar bubbles as we show that any bubble that violates the bound is unstable. Since the original proofs are elementary but long, we chose to use an argument from [HMRR] to prove these results. We find a bound on the number of convex components and then from this bound we establish the first-ever bound for the number of components for the whole bubble. However the latter bound needs a condition that might be unnecessary as we discuss a possible extension in section 7.5. The bound on the number of convex components states that any minimizing m-bubble has at most m disjoint nonhexagonal convex components. This new bound leads to many impressive results besides this dissertation. It makes the study of stability in planar double bubbles [MW] possible. Also it is the key ingredient in the study of double bubbles on other flat surfaces such as the cylinder [CHLS1] and the torus [CHLS2]. However we found that it is not necessary in the study of double bubble in a corner [FW]. In chapter 5, we focus on the geometry of planar bubbles. Many local symmetries and the local uniqueness of shapes are found. The study of reduction and decoration by 3-sided components in section 5.2 helps us understand some hidden geometries of bubbles very well. In section 5.3, we develop a useful tool for eliminating many candidates by showing that there is no illegal pair of barbequed components in minimizing bubbles. At the end of this chapter, we prove symmetry for some key components and also for some key bubbles. In chapter 6, we prove the planar triple bubble conjecture using the weak approach. The convex component bound forces the highest pressure region to have at most 3 components. This, together with other basic results, leaves only four hundred potentially minimizing combinatorial types. The deep understanding on geometry of planar bubbles helps narrowing down to only 54 nonstandard potential combinatorial types. In section 6.3, we eliminate all the 54 candidates by geometric information from chapter 5 and by the variational argument from chapter 4. This finishes the proof. The last chapter discusses related topics and progress on the planar soap bubble problem for more than three regions. We found a naturally beautiful conjectural isoperimetric inequality. After a exhaustive work, we found a convincing way to prove the triple bubble conjecture using simply overlapping bubbles. All of these seem to be a promising way to prove or at least narrow down the quadruple bubble problem and the general problem. 2

10 Chapter 2 Preliminary results In this chapter, we will summarize previously known results on the soap bubble problem. We focus on progress in the planar case. In the last section, we will see partial progress to the planar triple bubble conjecture. 2.1 Existence and regularity of planar soap bubbles The soap bubble problem for m volumes in R n is the search for the way to enclose and separate m regions of given volumes using least surface area. If n = 2, we call the problem the planar soap bubble problem. We say E R n is an enclosure of volumes v 1,..., v m in R n if E is closed and bounded with finite (n 1)-dimensional Hausdorff measure H n 1 and R n \ E contains open regions of volumes v 1,..., v m. For each enclosure E of volumes v 1,..., v m, let R i be the open region of volume v i. We call R n \ R 1... R m the exterior region R 0. Hence R 0 is the only unbounded region. Since E is equal to the boundary of R 0... R m and thus E is equivalent to the cluster of regions R 1,..., R m, we will also call E a cluster. Each connected component of a region is shortly called a component. A bounded component of R 0 is specifically called an empty chamber as it does not contribute volume to any of the m bounded regions. If E has least Haussdorf measure H n 1 among enclosures of the given volumes, we say E is minimizing. E is said to be standard if all regions are connected and if m n+1, every region must meet all other regions with structure described in Section 2.3. We conjecture that every minimizing cluster is standard and refer to this conjecture as the soap bubble conjecture. The existence and basic regularity of solutions to the soap bubble problem in R n for n 3 was proved by Almgren in 1976 [Almgren]. In 1987, Bleicher observed a number of necessary regularity conditions for minimizing clusters [Bleicher1]: Theorem 2.1. [Bleicher1] A minimizing cluster is a connected embedded trinary graph with circular or straight edges meeting at equal 120 angles. The sum of the oriented curvatures of the 3 edges at each vertex is zero. We call the condition from the last statement of the previous theorem, the cocycle condition at a vertex. In 1994, Morgan proved the existence and regularity of solutions to the planar soap bubble problem. Theorem 2.2. [Morgan] For A 1,..., A m > 0, there is a minimizing cluster of areas A 1,..., A m. Every minimizing enclosure is an embedded graph with finitely many circular or straight edges 3

11 meeting at equal 120 angles at vertices of degree 3. All edges separating a specific pair of regions have the same curvature. From now on we will focus on the planar soap bubble problem unless otherwise noted. A curve segment is said to be circular if it has constant curvature. Hence a straight segment is also considered circular. Corollary 2.3. [Bleicher1] A minimizing planar enclosure is path connected and thus each component is simply connected. Proof. This proof is from the proof of Theorem 2.1. If the minimizing enclosure is not path connected, by sliding 2 pieces of the enclosure until they touch, we create a cluster of the same length and areas but with an illegal meeting between edges. This contradicts the previous 2 theorems. Hence the enclosure is connected and thus every component is simply connected. Definition 2.4. For given v 1,..., v m > 0, let A(v 1,..., v m ) be the hyper-surface area H n 1 of the minimizing cluster enclosing volumes v 1,..., v m. Specifically for 2 dimensions, let L(A 1,..., A m ) be the length of the minimizing cluster enclosing areas A 1,..., A m. Proposition 2.5. [Hutchings] The mappings A and L are continuous. For a cluster with circular edges, we say p 1,..., p m R are pressures of R 1,..., R m if each edge between R i and R j has curvature p i p j and curves into the lower pressure region with the convention that p 0 = 0. Existence of pressures implies the cocycle condition at all vertices and further implies that all edges separating a specific pair of regions have the same curvature. An edge of a cluster is redundant if it separates a region from itself. Hence a cluster with redundant edges is clearly not minimizing. A cluster is called regular if it has no redundant edges, all its edges are circular and only meet in threes at 120 angles, and it has pressures for its regions. We also call a regular cluster a bubble. A bubble with 2, 3, 4, or m areas is called double bubble, triple bubble, quadruple bubble, or m-bubble, respectively. Montesinos proved that the shape of a standard m-bubble in R n where m n + 1 is uniquely determined by given m volumes [Montesinos]. 2.2 Results on minimizing bubbles We consider a cluster as an embedded graph in R 2 with label 1, 2,..., m or 0 on each face to tell what region a component contributes area to, and when we consider a continuous deformation of a cluster, we fix the label on each face. Note that the unbounded face is always labeled 0. We follow [CHHKLMT] in defining variations and pressures. Definition 2.6. Let G R 2 be an embedded graph. We say a cluster C has a combinatorial type G if there is a continuous deformation f t of G such that f 0 is the identity, f t is injective, and f 1 (G) = C. Definition 2.7. A variation of a bubble B is a C 1 family of clusters {B t } t <ε of combinatorial type B, with B 0 = B. Let l(b) denote the length of B, which is the total length of the edges of B. 4

12 Proposition 2.8. [CHHKLMT] For a bubble B with areas A 1,..., A m and pressures p 1,..., p m, and any variation {B t } of B, we have dl(b t ) dt = 0 m i=1 p i da i (t) dt where A i (t) denotes the area of the i th bounded region of B t. Proof. Let E ij be the union of all edges between R i and R j, and u ij the size of the normal component of the initial velocity of the variation on E ij from R j into R i. By Corollary 4.4, dl(b t )/dt 0 = i<j (p i p j ) E ij u ij. Since E ij u ij is the initial rate of decrease in the area of R i taken by R j, we have j E ij u ij = da i /dt. Let a ij = E ij u ij. Since p 0 = 0, (p i p j )a ij = p i a ij p j a ij = p i a ij p i a ji = p i a ij p i a ji i<j i<j i<j i<j j<i i>0 i<j i>0 j<i = ( p i a ij + i>0 j>i j<i Therefore we have the desired equation. p i a ij ) = i>0, 0 p i a ij = p i da i /dt 0. i>0 An edge e is said to be incident to a vertex v if v is an endpoint of e. Proposition 2.9. [CHHKLMT] [Bleicher3] For a closed path that crosses only edges of a minimizing cluster, the sum of the oriented curvatures of the edges crossed is zero. Proof. This proof is from [Bleicher3]. The conventional sign of an edge is positive if we cross into the convex side, and is negative otherwise. Let B be a minimizing cluster and let γ be a directed Jordan curve that does not cross B at any vertex and crosses edges for finitely many times. Let v i be the vertices of B inside γ and let γ i be directed Jordan curves around v i with the same orientation as γ such that each γ i crosses B only at the 3 incident edges of v i and nowhere else. By Theorem 2.1, the sum of the oriented curvatures of the 3 edges on each v i is zero. Then the sum of the oriented curvatures of the edges crossed by γ is equal to the sum over v i of the sum of the oriented curvatures of the 3 edges crossed by γ i, which is a sum of zero and hence is equal to zero as wanted. Remark The previous proposition is obviously true for a bubble since the oriented curvature of each crossed edge is the pressure change. We can easily see that going around components and getting back to the original component simply results in a zero total change of pressure. Proposition [CHHKLMT] A minimizing cluster is regular. Proof. By Theorem 2.1 or Theorem 2.2, we have to show only that each region has a pressure. In the first step, we will regard each component as a new region and find a pressure for it. By the previous proposition, we can assign a pressure for each component C to be the sum of the oriented curvatures of the edges crossed by a chosen path from the exterior component into C. Note that the exterior component gets pressure 0. If 2 components of a region get different pressures p < q, by the previous proposition, we may move some area from the higher pressure one to the lower one. This variation initially decreases length by rate q p per unit area. Hence we can make a shorter cluster with the same original areas. Therefore each region has a well-defined pressure. j i 5

13 Corollary [CHHKLMT] For a bubble B and an area-preserving variation {B t } of B, we have dl(b t )/dt 0 = 0. Proof. Since da i /dt 0 = 0, Proposition 2.8 implies the desired equation. Proposition [CHHKLMT] A bubble of areas A 1,..., A m and pressures p 1,..., p m has total length 2 p i A i. Proof. Consider a variation dilating the bubble with ratio 1 + t around the origin. Hence l(b t ) = (1 + t)l(b) and A i (t) = (1 + t) 2 A i. Therefore l(b) = dl(b t )/dt 0 = p i da i /dt 0 = 2 p i A i Remark Note that some regions may have negative pressure, but from the previous proposition, we can easily see that the highest pressure must be positive. Definition The sign of the curvature of a directed edge is considered positive[negative] if the edge is turning left[right]. Remark When considering a component C, we implicitly direct its edges counter-clockwise with C thus to the left on each edge. Hence the signed curvature of an edge of a component is well-defined. Definition The turning angle of an edge of a component is the product of its signed curvature and its length. Lemma For an n-sided component of a bubble, the sum of all edges turning angles is 6 n if the component is bounded and is 6 n 3 π if the component is unbounded. Proof. This is clear since the total curvature of the component, as a simple closed curve, is 2π, and since the external angle at each corner is π 3. Corollary If a 3-sided component of a bubble meets 2 higher-pressure components, then it must have an edge of turning angle greater than π. Proof. For a given 3-sided component, the sum of the 3 turning angles is π by the previous lemma. Since the 2 edges that meet higher-pressure components have negative turning angles, the turning angle of the other edge is greater than π. Remark The previous corollary is also valid if one of the 2 surrounding components has the same pressure as the 3-sided component. Proposition [Bleicher3] For a minimizing bubble, any 2 components may meet at most once, along a single edge. Proof. Suppose 2 components C and D meet along 2 edges e and f. If we remove e and f, we can see that C and D enclose a subcluster. We will use cut and paste technique to create an irregular cluster. First cut e and f along 2 dotted lines as in the figure. Next, flip the 3-sided piece and glue it back as in the figure. Since the new cluster has the same length and areas as the original one, it is minimizing. If the new cluster has a broken edge, it is irregular. If not, we can increase the size of the 3-sided piece. Then, eventually, the subcluster will touch another part of the cluster and create an illegal meeting. An incident edge of a component C is an incident edge at a vertex of C that is not an edge on the boundary of C. 6 3 π

14 Figure 2.1: We can cut a wedge, flip and glue it back to get an irregular cluster. Corollary A minimizing m-bubble has no 2-sided component if m 3. Proof. If there is a 2-sided component, then the 2 components surrounding the 2-sided component meet twice unless the 2 incident edges on the 2-sided component are the same edge. By the previous proposition, the latter case is true. The 2-sided component plus this third edge form a standard double bubble. Since m 3, there is another bounded component not attached to this part, contradicting Corollary 2.3. Lemma For a minimizing bubble, any 2 regions may not share 2 edges of turning angles greater than π. Proof. Suppose there is such a pair of edges. First we cut out from the two edges arcs of turning angles π ε and π + ε, for some small ε. Next we switch these 2 arcs and glue them back. For small enough ε, this produces a new cluster of the same combinatorial type. Then we have a new cluster with the same length and areas as before, but this cluster is irregular. This contradicts the minimality of the original bubble. Remark The previous lemma is also valid if one of the edges has turning angle exactly π. 2.3 Geometry of standard bubbles We discuss geometry of standard bubbles and focus on the planar case of few areas. In [SM], Sullivan gave the structure of the standard m-bubble in R n for m n + 1 and conjectured that the structure is unique in many senses. The way to construct the bubble starts with a regular m-simplex in S m 1. Next lift this to S n along longitudes. Finally stereographically project this into R n. The result bubble has S n m symmetry. Sullivan also conjectured that the standard m-bubble is unique for given m volumes and is uniquely minimizing. Later Montesinos proved part of these conjectures. Theorem [Montesinos] For any given m volumes, there is a unique standard m-bubble enclosing those m volumes in R n where m n + 1. For planar case, a standard double bubble is composed of 3 circular edges and 2 vertices as in Figure 2.2. It was proven in [FABHZ] that the standard double bubble is the unique minimizing cluster for given 2 areas. A standard triple bubble is composed of 6 circular edges and 4 vertices as in Figure 2.3. The standard triple bubble was proved [CHHKLMT] to be shortest among clusters of connected regions. 7

15 Figure 2.2: A standard double bubble (a figure from [RHLS]). Figure 2.3: A standard triple bubble (a figure from [Montesinos]). 2.4 Weak approach to the planar triple bubble conjecture In the proof of the planar double bubble conjecture [FABHZ], a new approach was introduced in order to rule out existence of empty chambers. The authors proved that a minimizing double bubble has no empty chambers and then proved that a minimizing double bubble without empty chambers is standard. To see briefly how this approach works, consider a bubble with an empty chamber. If we assign the chamber to be part of one of the neighboring components and then remove the redundant edge, we would get a shorter cluster that encloses bigger areas. Later, Vaughn showed [Vaughn] explicitly how this idea works for the problem of 3 areas. Lemma Let A be the least area function of Definition 2.4 and fix V 1,..., V m > 0. Then the minimum min vi V i A(v 1,..., v m ) is attained. Proof. This proof is based on [FABHZ] and [Vaughn]. For each i, A(v 1,..., v m ) is greater than the area of the sphere of volume v i. Therefore lim vi A(v 1,..., v m ) =. Hence the continuous function A takes its minimum in [V 1, )... [V m, ). Definition The clusters that attain the minimum surface area as in the previous lemma are called weakly minimizing bubbles or weak minimizers for the given volumes V 1,..., V m. Remark By the first paragraph of this section, a weak minimizer has no empty chambers. By definition, each weak minimizer is a minimizing cluster for the volumes v i that it encloses. According to Figures 2.2 and 2.3, we can intuitively see the following 2 lemmas. proved from explicit trigonometric formulas for the lengths. They are Lemma [FABHZ] The length of a planar standard double bubble is strictly increasing in each of the two areas. 8

16 Lemma [Vaughn] The length of a planar standard triple bubble is strictly increasing in each of the three areas. Theorem [Vaughn] The planar triple bubble conjecture holds if every weakly minimizing triple bubble is standard. Proof. Let B be a minimizing triple bubble of areas A 1, A 2, A 3. Suppose B is not a weak minimizer. Then B is longer than a weak minimizer M, which is standard by the assumption. By the previous lemma, M is no shorter than a standard triple bubble T of areas A 1, A 2, A 3. Hence B is longer than T, a contradiction. Thus B is a weak minimizer. Therefore B is standard by the assumption. So the planar triple bubble conjecture holds. Remark This theorem suggests an approach to proving the planar triple bubble conjecture, which we refer to as the weak approach. In section 3.2, we will prove that the weak approach is also valid for the planar bubble problems of m 6 areas. Vaughn has proven the equal pressure case for the weak approach in his dissertation. Theorem [Vaughn] A minimizing triple bubble with equal pressures and without empty chambers is standard and has equal areas. Although a standard triple bubble has equal area if and only if it has equal pressure, the previous theorem does not settle the planar triple bubble conjecture for equal areas. However, it does rule out a certain case of bubbles. Hence, by the weak approach, we have to rule out only triple bubbles without empty chambers and with unequal pressures. 9

17 Chapter 3 Basic results Here we describe some new basic results on planar bubbles. 3.1 Conditions on bubbles This section shows some simple results on planar bubbles that, in fact, are also valid in higher dimensions. Proposition 3.1. Let C be a minimizing bubble of areas A 1,..., A m. Suppose we can repartition the components of C in m + 1 groups such that the total area of the i th group is A i, 1 i m. Then the relabeled cluster still has no redundant edges and still has a consistent choice of pressures p i such that each edge between groups i and j has curvature p i p j. Proof. Consider the relabeled cluster. Since it has the same length and areas as the original, it is also minimizing. Therefore there are no redundant edges and there are consistent pressures. Example 3.2. By the previous proposition, a minimizing bubble may not have a component of nonzero pressure with the same area as an empty chamber: otherwise, we could switch these 2 components and the new cluster would have an empty chamber with nonzero pressure, a contradiction. Since each enclosure is an embedded planar graph, it has a dual graph. The distance between 2 components is the distance in the dual graph of the cluster. We say 2 components are n steps away from each other if the distance between them is n. A component is convex if its edges have nonnegative curvatures. Proposition 3.3. Every component of a bubble is within N 1 steps away from a convex component where q 1 > q 2 >... > q N are distinct pressures of all regions. In particular, a component of pressure q i is within just i 1 steps from a convex component. Proof. If the component does not have a higher-pressure neighbor, then it is convex itself. If it has such a neighbor, we can repeat tracking the neighbor and would find a convex component within i 1 steps in total. The following 2 lemmas do not yet have analogous results in higher dimensions. The next lemma discusses a well-known belief that a small component must have a high pressure compared to its neighbors. 10

18 Lemma 3.4. For n 6, an n-sided component with perimeter l of a bubble has an edge of curvature at least 6 n 3l π if n < 6 or curvature at most n 6 3l π if n > 6. P ki l i Proof. Consider a 3-sided component with edge lengths l 1, l 2, l 3 and signed curvatures k 1, k 2, k 3. By Lemma 2.18, k i l i = π. Hence max k i P li = P π li. Similarly for a 4-sided component, max k i P2π 3 l i and for a 5-sided component, max k i Pπ 3 l i. For an n-sided component with n > 6, k i l i = 6 n 3 π. Hence min k i n 6 P 3 l i π < 0. The next lemma tells that for a given bubble, non-6-sided components are not arbitrarily small in perimeter. In particular, their perimeters are bounded below. Lemma 3.5. For a bubble with pressures p i, let k := max p i min p i, the maximum pressure difference. Then the perimeter of any n-sided component is at least n 6 3k π. Proof. From the proof of the previous lemma, a 3-sided component with edge curvatures k 1, k 2, k 3, π has perimeter at least max k i π k. Similarly, a 4-sided component and a 5-sided component have perimeters at least 2π 3k and π 3k. For an n-sided component with n > 6, its perimeter at least than n k π. It is trivial that a 6-sided component has perimeter greater than 0 = 3k π. 3.2 The weak approach In this section, we extend the results from section 2.4 to show that the weak approach is also valid for the planar m-bubble problem with m 6 and for the m-bubble problem in R n with m n + 1. In the last chapter, we will discuss the possibility of using the weak approach for planar m-bubbles with m 7. Proposition 3.6. A weak minimizer for areas A 1,..., A m has no empty chamber. Its pressures p 1,..., p m are nonnegative. Each region R i has area exactly A i unless p i = 0. Proof. Let W be a weak minimizer. If W has an empty chamber, we can reassign it to be part of one of the neighboring components and then remove the redundant edge. Then we get a shorter cluster, contradicting the minimality of W. If p i < 0, as we increase the area of R i, the length decreases by Proposition 2.8, again giving a contradiction. If the area of R i is greater than A i and p i > 0, as we decrease the area of R i, length again decreases. Hence we get a shorter cluster that still contains areas bigger than A 1,..., A m, a contradiction. Remark 3.7. The previous proposition is also valid in higher dimensions. Theorem 3.8. For m 6, the planar m-bubble conjecture holds if every weak minimizer is standard. Proof. Let W be a weak minimizer for areas A 1,..., A m ; it is standard by the assumption meaning that the regions are connected. First we show that W has areas A 1,..., A m. Suppose region R i has area greater than A i. Hence p i = 0 which is the lowest pressure, so the turning angle for each edge of R i is nonpositive. Since m 6 and regions are connected, R i has at most 6 sides, and then by Lemma 2.18, R i is hexagonal. Hence m = 6 and R i touches all other regions including R 0. Thus all neighboring regions have pressure 0, which contradicts Remark So W has areas A 1,..., A m, and is therefore a minimizer. Now let M be any minimizer. Since l(m) = l(w ), M is also a weak minimizer. By the assumption, M is standard. 11

19 Remark 3.9. This theorem shows that, to prove the planar soap bubble conjecture for m 6, it suffices to consider nonstandard clusters without empty chambers and with nonnegative pressures, and to show they are not weakly minimizing. Theorem The m-bubble conjecture in R n, where m n+1, holds if every weakly minimizing m-bubble is standard. Proof. Let V 1,..., V m be prescribed volumes. Assume that every weakly minimizing m-bubble is standard. Let W be a weak minimizer. By the assumption, W is standard. By [Montesinos], W has positive pressures since every bounded component meets the exterior region through a strictly convex, spherical surface. By Proposition 3.6 and the remark following it, W has volumes V 1,..., V m, so W is a minimizer. Let M be a minimizer. Since M has the same surface area as W, the cluster M is also a weak minimizer. By assumption, M is standard. Therefore, the m-bubble conjecture holds. 3.3 Using the weak approach Here we show some consequences of the weak approach. Note that the component bounds here are usually not as good as those in Chapter 4. Lemma In a weakly minimizing bubble, let C 1,..., C n be components of a single region with areas a 1,..., a n, and suppose each C i meets some region S i along edges of total length l i. Then li < 2 π a i. Proof. Suppose l i 2 π a i. We will make a shorter cluster with greater areas. First, for each i, we reassign C i to be part of S i and then remove the edges of length l i. Next add a circle of area a i to be part of the new cluster. Now we have a new cluster with length the original length plus 2 π a i l i. Since l i 2 π a i, the new cluster is no longer than the original cluster. Since the new cluster has disconnected enclosure, this contradicts Corollary 2.3. Therefore li < 2 π a i. Lemma An n-sided component of area a of a weakly minimizing bubble has perimeter less than 2n πa. Proof. Suppose a component has perimeter at least 2n πa. Then it has a side of length at least 2 πa. This contradicts Lemma The next proposition tells that a non-6-sided component of a weakly minimizing bubble may not be arbitrarily small in area. Proposition In a weakly minimizing bubble with maximum pressure difference k, the area of any non-6-sided component is at least π 1764k 2. Proof. Consider a component C of area a and of n 6 sides. By Lemma 3.5, the perimeter of C is greater than n 6 n 6 3k π. By the previous lemma, 3k π < 2n πa. Hence a > n 6 6nk π > π 42k. Therefore a > π 1764k 2. Proposition For a weakly minimizing triple bubble, no region may have 18 components for which the ratio of the largest area to the smallest area is at most 2. 12

20 Proof. Suppose there are 18 components C 1,..., C 18 of a region R 1 with areas a 1 a 2... a 18 where a 18 a 1 2. Then 1 ai ( a a 1 ) 1 3 ( ) a 18 > a18 17a 18 + a 1 ai. Since each C i has perimeter greater than 2 πa i, it must meet one of the regions R 2, R 3, or R 0 along edges of total length greater than 2 3 πai. By the previous lemma, 2 3 πai < 2 π a i, which contradicts the calculation above. Remark We could generalize the previous proposition as follows. By the same calculation, if 1 3 (1 + n α ) n + 1 α, then a region has at most n + 1 components such that the ratio of the largest area to the smallest area is at most α. Lemma Suppose we have a nondecreasing sequence 0 < x 1 x 2... x N. If, for some α and n, we have x n+i x x i α for all i, then N n(log N α x1 + 1). Proof. Suppose x n+i x i α for all i. Then x n+1 x 1, x 2n+1 x n+1,... α. Thus So log α x N x1 x N = x n+1 x 2n+1 x N 1 n... n+1 x 1 x 1 x n+1 x ( N 1 n 1)n+1 x N 1 n x N N 1 n > N 1 n 1, and hence N n(log α x N x1 + 1). α N 1 n. n+1 Proposition For a region R of a weakly minimizing triple bubble, let a max and a min be the areas of the biggest and the smallest components. Then R has at most 17(log 2 a max a min +1) components. Proof. Let a 1 a 2... a N be the areas of the components of R. By Proposition 3.14, a 17+i a i 2 a for all i. By the previous lemma, N 17(log max 2 a min + 1). Remark A similar bound holds, with different coefficients, for any bubble problem where the weak approach is valid. 13

21 Chapter 4 Bounds on the number of components One of the few most important things on the way to prove the soap bubble problem is knowing the bound of the number of components. Bounds for the number of components for minimizing double bubbles in R n have been proved for only small n [HLRS]. Unfortunately there is no previously known bound for a minimizing planar triple bubble even for the case of equal areas. In this chapter, we will find a bound for the number of nonhexagonal convex components for planar bubbles. From this bound, we will establish the first bound for the total number of components for planar bubbles. 4.1 Examples of a nonminimizing bubble We will show 2 examples of nonminimizing bubbles that have 2 identical twin components. The second example shows an elementary technique that, indeed, can be generalized to prove all the results in this chapter. Example 4.1. Suppose that a region R has 2 identical twin 3-sided components C 1 and C 2 surrounded by components D 1, E 1, F 1 and D 2, E 2, F 2 respectively (see Figure 4.1) where D 1 and D 2 belong to a region S, E 1 and E 2 belong to a region T, and F 1 and F 2 belong to a region U. Suppose the pressure difference between R and S is 1 and that S, T and U have equal pressure. Hence the curvatures of the edges of C i are 1 and the curvatures of the 3 incident edges of C i are 0. Thus, for each component C i, by Lemma 5.10, its 3 edges are isometric and 3 incident edges are straight pointing to the center of C i. To make a shorter cluster, simply scale C 1 down around its center so that each of its 3 vertices moves inward for a distance ε, and scale C 2 up so that each of its 3 vertices moves outward for a distance δ. So these 2 scalings keep 6 incident edges straight and dilate with scales 1 3ε and 1 3δ. the number δ is chosen so that the area of R is fixed. Hence the areas of S, T and U are also fixed. To maintain the area of R, we must have (1 3ε) 2 + (1 + 3δ) 2 = 2. Hence (1 + 3δ) 2 = ε 3ε 2 < ε + 3ε 2 = (1 + 3ε) 2, so δ < ε. By a simple calculation, the length of each original circular edge is π 3 and the change in length of the edges around the shrinking C 1 is 3 π 3 (1 3ε) + 3ε 3 π 3 = (3 3π)ε. Similarly, for the enlarging C 2, the change in the length around it is 3 π 3 (1 + 3δ) 3δ 3 π 3 = ( 3π 3)δ. Thus the change of total length is (3 3π)ε + ( 3π 3)δ = ( 3π 3)(δ ε) < 0. Therefore we get a shorter cluster. We define the base angle of a circular arc to be the angle between the arc and the segment joining its 2 endpoints. In example 4.1, the base angles remain constant at

22 Figure 4.1: Two identical twin 3-sided components each with 3 equal pressure surrounding components Figure 4.2: An arc of smaller base angle has correct area Example 4.2. Using the cluster from the previous example, we can also make a shorter cluster by shrinking C 1 moving the vertices distance ε as before. For C 2, move its 3 vertices outward along the 3 incident edges for distance ε and complete the new component C 2 with 3 edges each of base angle θ chosen so that the area of R is fixed. Hence the areas of S, T and U are also fixed. By a simple calculation, the change in length of the edges around C 1 is (3 3π)ε. If the 3 edges of C 2 uses base angle 30, the area of R would increase and the change in total length would be (3 3π)ε + ( 3π 3)ε = 0. Hence θ < 30, and thus the change in total length is negative as desired. 4.2 Variations of bubbles In this section, we will discuss variations of planar bubbles following [HMRR] and [MW]. We will find a formula for the second derivative of the perimeter of a bubble. Consider a planar cluster B with smooth interfaces E ij between R i and R j. Let N ij be the unit normal vector on E ij from R j into R i. Consider a continuous variation V = {B t : B R 2 } t <ε of B that is smooth on each E ij up to the boundary. The associated vector field is X := db t /dt 0. The scalar normal component of X from R j into R i is u ij := X N ij. Let k ij be the curvature of E ij ; this is nonnegative if R i has higher-pressure. Note that N ij, u ij, k ij are skew-symmetric in their indices. Let N, u, k be the disjoint union functions i<j N ij, i<j u ij, i<j k ij. Note that at a vertex, the three values of N, u, k are not equal. The normal component of X is un, where un denotes the pointwise product of the functions u and N. Given a scalar or vector valued function f = f ij defined on the interfaces of B, we define a function Y (f) on the vertices of B by Y (f)(p) = f ij (p) + f jh (p) + f hi (p) if R i, R j, R h meet at p (in that order counterclockwise). If Y (f)(p) = 0, we say f or f ij agree at p. For a bubble, since N ij agree at p for any X and the 15

23 associated normal component u, Y (u)(p) = X Y (N)(p) = 0. Hence u ij agree at p. Initially the area of R i changes at the rate j E ji u ji. We say V has character (x 0,..., x m ) R m+1 if the areas the of R i initially change with rates x i. We also say skew-symmetric real-valued functions ũ ij on E ij have character (x 0,..., x m ) if j E ji ũ ji = x i. If u ij is the scalar normal component of V, then the character of u ij is equal to the character of V. We say V is steady if each area of B changes at a constant rate Second variation formula We will calculate the second variation formula after the first variation formula. We let T (p) be the sum of the unit tangent vectors to the edges meeting at p. Note that T (p) = Y (N)(p). Lemma 4.3. (First variation of length for a planar cluster) For a cluster B with smooth interfaces E ij and a variation with associated vector field X and scalar normal component u ij, the initial first derivative of length is k ij u ij X(p) T (p) = ku X(p) T (p). i<j E ij vertex p B vertex p Proof. Let V be a variation with associated vector field X. We will first find the first variation of the length of a single edge. Let e be an edge of length l 0 and γ 0 : [0, l 0 ] R 2 be the parametrization of e by the arclength parameter s. For each t ( ε, ε), let γ t : [0, l 0 ] R 2 be the corresponding parametrization of the deformed edge e t of e at time t under V. Note that this is not usually an arclength parametrization of e t. Let E and N be the unit tangent vector and the unit normal vector of γ 0 where N is the 90 -rotation version of E (the north and east directions have the same relation). We denote the partial derivatives f t and f s of any function f by f and f. Hence γ 0 = E, γ 0 = X, and γ 0 = E = kn. Let l t be the length of e t. We will use O(t 2 ) to indicate a term that is in the order of t 2 and use that 1 + h = 1 + h 2 + O(h2 ). Hence dγ t l t = ds γ ds = 0 γ 0 + 2tγ 0 γ 0 + O(t2 )ds = (1 + tγ 0 γ 0 + O(t 2 ))ds = l 0 + t γ 0 γ 0 ds + O(t 2 ). By integration by parts, and remembering that u = N X, we have dl t dt ds = γ 0 γ 0 ds = γ 0 γ 0 l 0 0 γ 0 γ 0 ds 0 = E X l 0 0 kn X = ku E X s=0 + E X s=l0. By adding the first variation of every edge, we have the first variation formula, dl(b t ) dt = k ij u ij X(p) T (p) t=0 i<j E ij vertex p 16

24 Corollary 4.4. For a bubble with interfaces E ij, any variation has initial first derivative of length i<j k ij E ij u ij. Proof. This is clear since edges have constant curvatures and the sum of the 3 unit tangent vectors of edges is zero at every point. Remark 4.5. The formula in Lemma 4.3 is valid for any B t0 under a shifted variation {B t0 +t}. For smooth scalar normal components u and v, we define a symmetric bilinear form Q(u, v) = (u v k 2 uv) Y (quv) B vertex p where u is the derivative of u with respect to arc length along edges and q ij = k ih+k jh 3 p where R i, R j, R h meet. We write Q(u) for Q(u, u). at a vertex Lemma 4.6. Given a C 2 variation on a bubble with initial velocity X and normal component u, the first variation of curvature is dk dt = u + k 2 u. 0 Proof. We will verify the equality on each edge. We define e t, γ t, E, N, O(t 2 ) as in Lemma 4.3. Note that k t = γ t γ t. Let a be a function on e such that X = un + ae. Hence γ γ t t = γ 0 + t(un + ae) + 3 O(t 2 ). We have γ t = γ 0 + t(u N + un + a E + ae ) + O(t 2 ) = (1 + t( ku + a ))E + t(u + ka)n + O(t 2 ). Hence γ 3 = (1 + t( ku + a )) 2 + O(t 2 ) 3 = 1 + 3t( ku + a ) + O(t 2 ) and t γ t = t( k u ku + a )E + (1 + t( ku + a ))kn + t(u + k a + ka )N t(u + ka)ke + O(t 2 ) = t( k u 2ku + a + k 2 u)e + (k + t( k 2 u + 2ka ) + u + k a)n + O(t 2 ). Thus, as e has constant k, γ t γ = (1 + t( ku + a ))(k + t( k 2 u + 2ka + u )) + O(t 2 ) = k + t( 2k 2 u + 3ka + u ) + O(t 2 ) t and then γ t γ t = k + 3t( k2 u + ka ) + t(u + k 2 u) + O(t 2 ) γ t t( ku + a ) + O(t 2 = k + t(u + k 2 u) + O(t 2 ). ) Therefore we get the desired equality. Lemma 4.7. [HMRR] For a variation on a bubble, at a vertex p, we have dt (p) dt = Y ((qu + u )N). 0 17

25 Proposition 4.8. (Second variation of length for planar bubble) For a bubble with interfaces E ij and a steady variation, the initial second derivative of length is Q(u). Proof. This proof is based on the Proposition 3.3 of [HMRR] for a double bubble in R n. We need some adjustments to make it valid for a planar m-bubble. Let Eij t be the deformed E ij, let k t be the curvature of the deformed cluster, and let u t be the scalar normal component of X t = db t /dt t. Using the previous lemma, we can compute the derivative of the first term from Lemma 4.3 as follows, d dt k t=0 B t u t = u(u + k 2 u) + d k ij t B dt u t ij. i<j t=0 Eij t The latter sum is zero as can be seen as follows. Let a ij = d dt 0 E u t ij t ij be the second derivative of the area moved from R j into R i. Letting p i be the pressure of R i, normalized so that p 0 = 0, we have k ij a ij = (p i p j )a ij = p i a ij p i a ji = p i a ij. i<j i<j i<j j<i i>0 j i This is zero because for each i, j i a ij = d/dt 0 j E u t ij t ij which is zero because the variation is steady. Next, we compute the derivative of the second term from Lemma 4.3 using the previous lemma and that T 0 = 0. At a point p where R i, R j, R h meet, we have d/dt 0 (X t T t ) = X 0 Y ((qu + u )N) = Y (qu 2 + u u). We then have the second variation, using integration by parts, to be u(u + k 2 u) Y (qu 2 + u u) = B p B(u 2 k 2 u) Y (qu 2 ) = Q(u). p Remark 4.9. The previous lemma and proposition are also valid in higher dimensions by the general second variation formula from [HMRR]. Remark Since each of V and X uniquely determine u, we will also write Q(V ) or Q(X) to refer to Q(u). A bubble is unstable if it has an area-preserving smooth variation with negative second variation. An unstable bubble is clearly not minimizing. Let u = {u ij : E ij R} such that u ij = u ji (that is, the u ij are skew-symmetric in their indices). We say u is admissible if u ij is in the Sobolev space W 1,2 (the set of all functions in L 2 with derivative also in L 2 ) on E ij and u ij agree at every vertex (u ij +u jh +u hi = 0 where R i, R j, R h meet). Hence every scalar normal component is an admissible function. Recall that the character of u is (x 0,..., x n ) where x i = j E ji u ji. Furthermore, if x i is zero, positive, or negative, we say u preserves, increases, or decreases the area of R i respectively. Indeed, we can see that the character is a linear map on admissible functions. Let F be the set of admissible functions with zero characters. Hence F is clearly a vector space over R. We extend the definition of Q for all admissible functions. For smooth u and v, by integration by parts and the fundamental theorem of calculus, we have Q(u, v) = (u + k 2 u)v Y ((qu + u )v). For piecewise C 2 functions u and v, we then have Q(u, v) = (u + k 2 u)v Y ((qu + u )v) [u v] where [u v] is the jump of u v and indeed we sum over discontinuities of u v other than vertices. Next we will show that Q(u + v) = Q(u) + Q(v) when the supports of u and v are almost disjoint. 18