arXiv:1801.09296v1 [math.FA] 28 Jan 2018 rn o 404,wieteatoswr nrsdnea the at 2017. residence of in semester were Fall the authors suppor during the work California, while upon Berkeley, based 1440140, also resea is No. 2020 material Grant Horizon This tha Union’s 637851). project European No a ment the of part under is (ERC) results Council these to leading research The [email protected]. 2 1 eateto ahmtc,Uiest fTxsa utn E Austin. at Texas of University Mathematics, of Department eateto ahmtc,Tcno salIsiueo T of Institute Israel - Technion Mathematics, of Department srlo 5]adBrl 7 n17,asrsta mn all among that asserts 1975, in [7] Borell and [52] Tsirelson calculus-of-variations. of method direct the en oe eicniuu ihrsetto respect with semi-continuous lower being with Let Introduction 1 isop unique the are tripod-clusters null-sets). that prove we Moreover, asinwihe counterpart: Gaussian-weighted o iest eg pnst ihpeeiesot boundary smooth piecewise with sets open (e.g. sets nice For h asinwihe Ecien eiee faBrlset Borel a of perimeter (Euclidean) Gaussian-weighted The st s rpdcutr hs nefcscnito thre of consist interfaces whose an tripod-cluster, a use to is decompose to way perimeter oegnrly if generally, More ( h lsia asinioeiercieult,establi inequality, isoperimetric Gaussian classical The eetbihteGusa obeBbl ojcue h le the Conjecture: Double-Bubble Gaussian the establish We γ n γ − = n 2 γ − ) h asinDul-ubeConjecture Double-Bubble Gaussian The n 1 dmninlpaea 120 at plane -dimensional P ( eoetesadr asinpoaiiymaueo Euclid on measure probability Gaussian standard the denote ∂U γ ( U ) ) se ..[2) h egtdperimeter weighted The [32]). e.g. (see, ∶ = H sup mne Milman Emanuel k eoe the denotes › S U γ ( n div R n ∶ = X notreclso rsrbd(oiie asinmeasure Gaussian (positive) prescribed of cells three into ( aur 0 2018 30, January 2 ⟨∇ − ○ π 1 k γ nls(omn rpdo Y hp nteplane). the in shape “Y” or tripod a (forming angles ) dmninlHudr esr,let measure, Hausdorff -dimensional k n 2 ,X W, Abstract ∶ e = − e S x − 1 2 S 1 W 2 ⟩) n o Neeman Joe and L dx ( dγ x 1 ) ( a eevdfnigfo h uoenResearch European the from funding received has t H = γ ∶ ∶ X ) k e . − ovrec,adtu t elwith well fits thus and convergence, e yteNtoa cec onainunder Foundation Science National the by ted ∈ W c n noainporme(rn agree- (grant programme innovation and rch ahmtclSine eerhIsiuein Institute Research Sciences Mathematical cnlg,Hia300 sal mi:emil- Email: Israel. 32000, Haifa echnology, C ( al [email protected]. mail: x afhprlnsmeigalong meeting half-hyperplanes e c ∞ ) hdidpnetyb Sudakov– by independently shed ( dx. R P rmti iiies(pto (up minimizers erimetric γ n ), U oe sets Borel ( ; U T P ⊂ R ) γ R s Gaussian-weighted ast n ( 2 a h datg of advantage the has n U ) , ) sdfie as: defined is S X a space ean skont agree to known is ≤ S U γ 1 in

k . R eoeits denote n ( R having n , S ⋅ S) : prescribed Gaussian measure γ U v 0, 1 , halfplanes minimize Gaussian-weighted ( ) = ∈ [ ] perimeter Pγ U (see also [16, 2, 5, 6, 39]). Later on, it was shown by Carlen–Kerce [11] (see also [17,( 39,) 34]), that up to γ-null sets, halfplanes are in fact the unique minimizers for the Gaussian isoperimetric inequality. In this work, we extend these classical results to the case of 3-clusters. A k-cluster n Ω Ω1,..., Ωk is a k-tuple of Borel subsets Ωi R called cells, such that Ωi are = ( ) n ⊂ k { } pairwise disjoint, P Ω for each i, and γ R 1 Ω 0. Note that the cells are γ i < ∞ ∖ ⋃i i = not required to be connected.( ) The total Gaussian( perimeter= ) of a cluster Ω is defined as:

1 k Pγ Ω ∶ Q Pγ Ωi . ( ) = 2 i 1 ( ) = The Gaussian measure of a cluster is defined as:

k 1 γ Ω ∶ γ Ω1 ,...,γ Ωk ∆( − ), ( ) = ( ( ) ( )) ∈ k 1 k k where ∆( − ) v R v 0 , v 1 denotes the k 1 -dimensional sim- ∶= ∈ ∶ i ≥ ∑i 1 i = − plex. The isoperimetric{ problem for k-clusters= } consists of identifying( ) those clusters Ω k 1 of prescribed Gaussian measure γ Ω v ∆( − ) which minimize the total Gaussian ( ) = ∈ perimeter Pγ Ω . ( ) Note that easy properties of the perimeter ensure that for a 2-cluster Ω, Pγ Ω ( ) = P Ω1 P Ω2 , and so the case k 2 corresponds to the classical isoperimetric setup γ = γ = when( testing) ( the) perimeter of a single set of prescribed measure. In analogy to the n classical unweighted isoperimetric inequality in Euclidean space R , ⋅ , in which the Euclidean ball minimizes (unweighted) perimeter among all sets of( prescribedS S) Lebesgue measure [9, 32], we will refer to the case k 2 as the “single-bubble” case (with the = bubble being Ω1 and having complement Ω2). Accordingly, the case k 3 is called the = “double-bubble” problem. See below for further motivation behind this terminology and related results. A natural conjecture is then the following: Conjecture (Gaussian Multi-Bubble Conjecture). For all k n 1, the least (Gaussian- ≤ + weighted) perimeter way to decompose Rn into k cells of prescribed (Gaussian) measure k 1 n v int ∆( − ) is by using the Voronoi cells of k equidistant points in R . ∈ n Recall that the Voronoi cells of x1,...,xk R are defined as: { } ⊂ n Ωi int x R min x xj x xi i 1,...,k , = ∈ ∶ j 1,...,k − = − = › = S S S S where int denotes the interior operation (to obtain pairwise disjoint cells). Indeed, when k 2, the Voronoi cells are precisely halfplanes, and the single-bubble conjecture holds = 2 by the classical Gaussian isoperimetric inequality. When k 3 and v int ∆( ), the con- = ∈ jectured isoperimetric minimizing clusters are tripods (also referred to as “Y’s”, “rotors” or “propellers” in the literature), whose interfaces are three half-hyperplanes meeting along an n − 2 -dimensional plane at 120○ angles (forming a tripod or “Y” shape in the ( ) 2 plane). When v ∂∆( ), the 3-cluster has (at least) one null-cell, reducing to the k 2 ∈ = case (and indeed two complementing half-planes may be seen as a degenerate tripod cluster, whose vertex is at infinity). Tripod clusters are the naturally conjectured minimizers in the double-bubble case, as their interfaces have constant Gaussian mean-curvature (being flat), and meet at 120○

2 angles, both of which are necessary conditions for any extremizer of (Gaussian) perimeter under a (Gaussian) measure constraint – see Section 4. Note that tripod clusters are also known to be the global extremizers in other optimization problems, such as the problem of maximizing the sum of the squared lengths of the Gaussian moments of the 3 cells in R (see Heilman–Jagannath–Naor [23]). On the other hand, for the Gaussian noise-stability question, halfplanes are known to maximize noise-stability in the single- bubble case [8, 42], but tripod-clusters are known to not always maximize noise-stability in the double-bubble case [24]. Our main result in this work is the following: Theorem 1.1 (Gaussian Double-Bubble Theorem). The Gaussian Double-Bubble Con- jecture (case k 3) holds true for all n 2. = ≥ In addition, we resolve the uniqueness question: Theorem 1.2 (Uniqueness of Minimizing Clusters). Up to γn-null sets, tripod clus- ters are the unique minimizers of Gaussian perimeter among all clusters of prescribed 2 Gaussian measure v int ∆( ). ∈ 1.1 Previously Known and Related Results The Gaussian Multi-Bubble Conjecture is known to experts. Presumably, its origins may be traced to an analogous problem of J. Sullivan from 1995 in the unweighted Euclidean setting [53, Problem 2], where the conjectured uniquely minimizing cluster (up to null- k sets) is a standard k-bubble – spherical caps bounding connected cells Ωi i 1 which are n n{ 1 } = obtained by taking the Voronoi cells of k equidistant points in S R + , and applying ⊂ all stereographic projections to Rn. To put our results into appropriate context, let us go over some related results in three settings: the unweighted Euclidean setting Rn, on the n-sphere Sn endowed with its canonical Riemannian metric and measure (normalized for convenience to be a prob- ability measure), and finally in our Gaussian-weighted Euclidean setting Gn. Further results may be found in F. Morgan’s excellent book [40, Chapters 13,14,18,19]. • The classical isoperimetric inequality in the unweighted Euclidean setting, going back (at least in dimension n 2 and perhaps n 3) to the ancient Greeks, and = = first proved rigorously by Schwarz in Rn (see [40, Chapter 13.2], [9, Subsection 10.4] and the references therein), states that the Euclidean ball is a single-bubble isoperimetric minimizer in Rn. It was shown by DeGiorgi [9, Theorem 14.3.1] that up to null-sets, it is uniquely minimizing perimeter. Long believed to be true, but appearing explicitly as a conjecture in an undergraduate thesis by J. Foisy in 1991 [18], the double-bubble case was considered in the 1990’s by various authors [19, 22], culminating in the work of Hutchings–Morgan–Ritor´e–Ros [26], who proved that up to null-sets, the standard double-bubble is uniquely perimeter minimizing in 3 R ; this was later extended to Rn in [46, 45]. That the standard triple-bubble is 2 uniquely perimeter minimizing in R was proved by Wichiramala in [57]. • On Sn, it was shown by P. L´evy [31] and Schmidt [50] that geodesic balls are single-bubble isoperimetric minimizers. Uniqueness up to null-sets was established by DeGiorgi [9, Theorem 14.3.1]. As in the unweighted Euclidean setting, the multi- bubble conjecture on Sn asserts that the uniquely perimeter minimizing k-cluster (up to null-sets) is a standard k-bubble: spherical caps bounding connected cells k Ωi i 1 and having incidence structure as in the Euclidean and Gaussian cases, { } =

3 2 already described above. The double-bubble conjecture was resolved on S by Masters [33], but on Sn for n 3 only partial results are known [15, 14, 13]. In ≥ particular, we mention a result by Corneli-et-al [13], which confirms the double- n 2 bubble conjecture on S for all n 3 when the prescribed measure v ∆( ) satisfies ≥ ∈ max v 1 3 0.04. Their proof employs a result of Cotton–Freeman [15] stating i i − ≤ that ifS the~ minimizingS cluster’s cells are known to be connected, then it must be the standard double-bubble. • We finally arrive to the Gaussian setting Gn which we consider in this work. As already mentioned, halfplanes are the unique single-bubble isoperimetric minimizers (up to null-sets). The original proofs by Sudakov–Tsirelson and independently Borell both deduced the single-bubble isoperimetric inequality on Gn from the one on SN , exploiting the classical fact that the projection onto a fixed n-dimensional subspace of the uniform measure on a rescaled sphere √NSN , converges to the n Gaussian measure γ as N → ∞. Building upon this idea, it was shown by Corneli- et-al [13] that verification of the double-bubble conjecture on SN for a sequence of 2 N’s tending to and a fixed v int ∆( ) will verify the double-bubble conjecture ∞ ∈ on Gn for the same v and for all n 2. As a consequence, they confirmed the n ≥ 2 double-bubble conjecture on G for all n 2 when the prescribed measure v ∆( ) ≥ ∈ satisfies max v 1 3 0.04. Note that this approximation argument precludes i i − ≤ any attemptsS to establish~ S uniqueness in the double-bubble conjecture, and to the best of our knowledge, no uniqueness in the double-bubble conjecture on Gn was 2 known for any v int ∆( ) prior to our Theorem 1.2. ∈ An essential ingredient in the proofs of the double-bubble results in Rn (and in many results on Sn), is a symmetrization argument due to B. White (see Foisy [18] and Hutchings [25]). However, it is not clear to us what type of Gaussian symmetrization will produce a tripod cluster (for a symmetrization which works in the single-bubble case, see Ehrhard [16]). A second essential ingredient in the above results is Hutchings’ theory [25] of bounds on the number of connected components comprising each cell of a minimizing cluster. In contrast, we do not use any of these ingredients in our approach. Furthermore, contrary to previous approaches, we do not directly obtain a lower bound on the perimeter 2 of a cluster having prescribed measure v ∆( ), nor do we identify the minimizing clusters ∈ by ruling out competitors. Our approach is based on obtaining a matrix-valued partial- differential inequality on the associated isoperimetric profile, concluding the desired lower 2 bound in one fell swoop for all v ∆( ) by an application of the maximum-principle. ∈ 1.2 Outline of Proof k 1 k 1 Let I( − ) ∆( − ) → R denote the Gaussian isoperimetric profile for k-clusters, defined ∶ + as: k 1 I( − ) v inf P Ω Ω is a k-cluster with γ Ω v . ∶= γ ∶ = ( ) { 2 ( ) 2 2 2( ) } 2 Our goal will be to show that I( ) Im( ) on int ∆( ), where Im( ) int ∆( ) → R denotes = ∶ + the Gaussian double-bubble model profile: 2 Im( ) v ∶ inf Pγ Ω ∶ Ω is a tripod-cluster with γ Ω v . ( ) = { ( ) ( ) = } 2 m m Note that for any v int ∆( ), there exists a tripod-cluster Ω with γ Ω v. Indeed, ∈ = by the product structure of the Gaussian measure, it is enough to establish( ) this in the

4 plane, and after fixing the orientation of the tripod, we actually show using a topological 2 2 argument that there exists a bijection between the vertex of the tripod in R and int ∆( ). 2 2 To establish I( ) Im( ), let us draw motivation from the single-bubble case. By 1 = 1 identifying ∆( ) with 0, 1 using the map ∆( ) v1, v2 ↦ v1 0, 1 , we will think of the [ 1 ] ∋ ( ) ∈ [ ] single-bubble profile I( ) as defined on 0, 1 . The single-bubble Gaussian isoperimetric 1 1 [ ] 1 inequality asserts that I( ) Im( ), where Im( ) 0, 1 → R denotes the single-bubble = ∶ + model profile: [ ]

1 Im( ) v ∶ inf Pγ U ∶ U is a halfplane with γ U v . ( ) = { ( ) ( ) = } 1 The product structure of the Gaussian measure implies that Im( ) may be calculated in dimension one, readily yielding:

1 1 Im( ) v ϕ ○ Φ− v , ( ) = ( ) 2 1 2 x 2 x where ϕ x 2π − ~ e− ~ is the one-dimensional Gaussian density, and Φ x ϕ y dy ( ) = ( ) ( ) = ∫−∞ (1 ) is its cumulative distribution function. It is well-known and immediate to check that Im( ) satisfies the following ODE:

1 1 I( ) ′′ on 0, 1 . (1.1) m = − 1 ( ) Im( ) [ ]

2 Our starting observation is that Im( ) satisfies a similar matrix-valued differential 2 2 equality on ∆( ). Let E denote the tangent space to ∆( ), which we identify with R3 3 x ∶ ∑i 1 xi 0 . Given A A12,A23,A13 with Aij 0, we introduce the following { ∈ = = } = ( ) ≥ 3 × 3 positive semi-definite matrix:

A12 + A13 −A12 −A13 T ⎛ ⎞ LA ∶ Q Aij ei − ej ei − ej −A12 A12 + A23 −A23 0. = 1 3 ( )( ) = ⎜ ⎟ ≥ i j A13 A23 A13 A23 ≤ < ≤ ⎝ − − + ⎠

In fact, as a quadratic form on E, it is easy to see that LA is strictly positive-definite as soon as at least two Aij ’s are positive. 2 m n 1 m m Given v int ∆( ), let A v γ − ∂Ω ∂Ω 0 denote the weighted areas of ∈ ij ∶= i ∩ j > the interfaces of a tripod cluster( ) Ωm satisfying( γ Ωm) v. A calculation then verifies = that: ( ) 2 2 1 2 ∇ Im( ) v −LA− m v on int ∆( ), (1.2) ( ) = ( ) where differentiation and inversion are both carried out on E. This constitutes the right extension of (1.1) to the double-bubble setting.

2 2 2 To establish that I( ) Im( ) on int ∆( ), our idea is as follows. First, it is not 2 = 2 hard to show that Im( ) may be extended continuously to the entire ∆( ) by setting 2 1 2 2 2 2 Im( ) v ∶ Im( ) maxi vi for v ∂∆( ). We clearly have I( ) Im( ) on ∆( ), with equality ( ) = ( ) ∈ ≤ 2 on the boundary by the single-bubble Gaussian isoperimetric inequality (where I( ) v 1 1 ( ) = I( ) maxi vi Im( ) maxi vi ). ( ) = ( ) 2 Assume for the sake of this sketch that I( ) is twice continuously differentiable on 2 2 int ∆( ). Given an isoperimetric minimizing cluster Ω with γ Ω v int ∆( ), let n 1 ( ) = ∈ Aij v ∶ γ − ∂∗Ωi ∩ ∂∗Ωj , i j, denote the weighted areas of the cluster’s interfaces, ( ) = ( ) < where ∂∗U denotes the reduced boundary of a Borel set U having finite perimeter (see Section 3). Assume again for simplicity that Aij v are well-defined, that is, depend ( )

5 only on v. It is easy to see that at least two Aij v must be positive, and hence LA v ( ) is positive-definite on E. We will then show that( the) following matrix-valued differential inequality holds: 2 2 1 2 ∇ I( ) v −LA− v on int ∆( ), (1.3) ( ) ≤ ( ) in the positive semi-definite sense on E. Consequently: 2 2 1 2 2 − tr ∇ I( ) v − tr LA v 2 Q Aij v 2I( ) v on int ∆( ). (1.4) [( ( )) ] ≤ ( ( )) = i j ( ) = ( ) < 2 On the other hand, by (1.2), we have equality above when I( ) and A are replaced by 2 m 2 2 2 I( ) and A , respectively. Since I( ) I( ) on ∆( ) with equality on the boundary, an m ≤ m application of the maximum principle for the (fully non-linear) second-order elliptic PDE 2 2 (1.4) yields the desired I( ) I( ). = m The bulk of this work is thus aimed at establishing a rigorous version of (1.3). To this end, we consider an isoperimetric minimizing cluster Ω, and perturb it using a flow Ft along a vector-field X. Since: 2 I( ) γ Ft Ω Pγ Ft Ω ( ( ( ))) ≤ ( ( )) with equality at t 0, we deduce (at least, conceptually) that the first variations must = coincide and that the second variations must obey the inequality. Such an idea in the single-bubble case is not new, and was notably used by Sternberg–Zumbrun in [51] to establish concavity of the isoperimetric profile for a convex domain in the unweighted Euclidean setting; see also [41, 29, 48, 4] for further extensions and applications, [28] for a first-order comparison theorem on Cartan-Hadamard manifolds, and [3] where the result- ing second-order ordinary differential inequality was integrated to recover the Gromov– L´evy isoperimetric inequality [20], [21, Appendix C]. On a technical level, it is crucial for us to apply this to a non-compactly supported vector-field X, and so we spend some time to revisit classical results regarding the first and second variations of (weighted) volume and perimeter. Contrary to the single-bubble setting, the interfaces Σ ∂∗Ω ∂∗Ω ij = i ∩ j will meet each other at common boundaries, which may contribute to these variations. Fortunately, for the first variation of perimeter, these contributions cancel out thanks to the isoperimetric stationarity of the cluster (without requiring us to use delicate regu- larity results regarding the structure of the boundary of interfaces – see Remark 4.10). For the second variation, we are not so fortunate, and in general the boundary of the interfaces will have a contribution. However, in simplifying our original argument for establishing (1.3), we obtain an interesting dichotomy: if the minimizing cluster is effectively one-dimensional (a case we do not a-priori rule out), it is easy to obtain (1.3) by a one-dimensional computation directly; otherwise, it turns out we obtain enough information from applying the above machinery to the constant vector-field X w in all directions w Rn, amounting to ≡ ∈ translation of the minimizing cluster. This simplifies considerably the resulting formulas for the second variation of (weighted) volume and perimeter, and allows us again to bypass the delicate regularity results mentioned before. Consequently, we only need to use the classical results from Geometric Measure Theory on the existence of isoperimetric minimizing clusters and regularity of their interfaces, due to Almgren [1] (see also Maggi’s excellent book [32]). To establish the uniqueness of minimizers, we observe that all of our inequalities in the derivation of (1.3) must have been equalities, and this already provides enough information for characterizing tripod-clusters.

6 The rest of this work is organized as follows. In Section 2, we construct the model tripod-clusters and associated model isoperimetric profile, and establish their properties. In Section 3 we recall relevant definitions and provide some preliminaries for the ensuing calculations. In Section 4 we deduce first-order properties of isoperimetric minimizing clusters (such as stationarity) in the weighted unbounded setting, as well as second-order stability. In Section 5, we calculate the second variations of measure and weighted- perimeter under translations. In Section 6 we obtain a rigorous version of the matrix- valued differential inequality (1.3) in both cases of the above-mentioned dichotomy. In Section 7, we conclude the proof of Theorem 1.1 by employing a maximum-principle. In Section 8, we establish Theorem 1.2 on the uniqueness of the isoperimetric minimizing clusters. Finally, in Section 9, we provide some concluding remarks on extensions and additional results which will appear in [35]. For brevity of notation, we omit the super- 2 2 2 2 script ( ) in all subsequent references to I( ), Im( ) and ∆( ) in this work. In addition, we use C 1, 2 , 2, 3 , 3, 1 to denote the set of positively oriented pairs in 1, 2, 3 . = {( ) ( ) ( )} { } Acknowledgement. We thank Francesco Maggi for pointing us towards a simpler version of the maximum principle than our original one, Frank Morgan for many helpful references, and Brian White for informing us of the reference to the recent [12]. We also acknowledge the hospitality of MSRI where part of this work was conducted.

2 Model Tripod Configurations

In this section, we construct the model tripod clusters which are conjectured to be 2 optimal on Rn. It will be enough to construct them on R , since by taking Cartesian n 2 product with R − and employing the product structure of the Gaussian measure, these clusters extend to Rn for all n 2. Actually, it will be convenient to construct them 2 ≥ 2 on E, rather than on R , where recall that E denotes the tangent space to ∆( ), which 3 3 2 we identify with x R x 0 . Consequently, in this section, let γ γ denote ∈ ∶ ∑i 1 i = = the standard two-dimensional{ = Gaussian} measure on E, and if Ψ denotes its (smooth, 1 1 Gaussian) density on E, set γ Ψ . ∶= H m Define Ωi int x E ∶ maxj xj xi (“m” stands for “model”). For any x E, m = m { ∈ m m = } ∈ x + Ω x + Ω1 ,x + Ω2 ,x + Ω3 is a cluster, which we call a “tripod” cluster (also = ( ) m m m referred to as a “Y” or “rotor” in the literature). Let Σij ∶ ∂Ωi ∩ ∂Ωj denote the m m = m interfaces of Ω . Observe that x + Σij , the interfaces of x + Ω , are flat, and meet at 120○ angles at the single point x. We denote:

m 1 m Aij x ∶ γ x + Σij . ( ) = ( )

We begin with some preparatory lemmas:

Lemma 2.1. For all distinct i, j, k 1, 2, 3 , let nij ej − ei √2 and tij ei + ej − ∈ { } = ( )~ = ( 2ek √6. Then nij and tij form an orthonormal basis of E, and: )~ m Aij x ϕ x, nij 1 − Φ x,tij (2.5) ( ) = (⟨ ⟩)( (⟨ ⟩)) m Proof. It is immediate to check that Σij can be parametrized (with unit speed) as atij ∶ { a 0 , and that tij and nij form an orthonormal basis of E. Consequently, for any a R, ≥ } ∈

7 2 2 2 atij + x a + x,tij + x, nij . Hence, S S = ( ⟨ ⟩) ⟨ ⟩ 2 1 1 ∞ atij x 2 γ x + Σij e−S + S ~ da ( ) = 2π S0 ∞ ϕ x, nij ϕ a + x,tij da = (⟨ ⟩) S0 ( ⟨ ⟩) ϕ x, nij 1 − Φ x,tij . = (⟨ ⟩)( (⟨ ⟩))

Lemma 2.2. For all x E and distinct i, j, k 1, 2, 3 : ∈ ∈ { } m xi − xj xi − xk γ x + Ωi min 1 − Φ xi , 1 − Φ , 1 − Φ , (2.6) ( ) ≤ Œ ( ) Œ √2 ‘ Œ √2 ‘‘

m xi − xj xi − xk γ x + Ωi 1 − Φ 1 − Φ . (2.7) ( ) ≥ Œ Œ √2 ‘‘ Œ Œ √2 ‘‘ Proof. For the upper bound, note that:

m x + Ωi int z E ∶ max zj − xj zi − xi z E ∶ zi − xi 0 , = { ∈ j ( ) = } ⊆ { ∈ > }

1 3 where the last inclusion follows since y z x E implies that max y 1 y 0. = − ∈ j j ≥ 3 ∑j j = Similarly, for any j i: = ≠ m x + Ωi int z E ∶ max zj − xj zi − xi z E ∶ zi − zj xi − xj . = { ∈ j ( ) = } ⊆ { ∈ > }

It remains to note that if Z is distributed according to γ on E, then Zi and Zi −Zj √2 are distributed according to the standard Gaussian measure on R, and (2.6)( follows.)~ For the lower bound, note that:

m x + Ωi z ∶ zi − zj xi − xj ∩ z ∶ zi − zk xi − xk . = { > } { > } By the Gaussian FKG inequality [44],

m γ x + Ωi γ z ∶ zi − zj xi − xj γ z ∶ zi − zk xi − xk , ( ) ≥ ({ > }) ({ > }) and (2.7) is established. Lemma 2.3. The map:

m E x ↦ V x ∶ γ x + Ω int ∆ ∋ ( ) = ( ) ∈ is a diffeomorphism between E and int ∆. Its differential is given by:

m m m m A12 + A31 −A12 −A31 1 ⎛ m m m m ⎞ 1 DV − −A12 A12 + A23 −A23 − LAm . (2.8) = √ ⎜ m m m m ⎟ = √ 2 A31 A23 A23 A31 2 ⎝ − − + ⎠

Proof. Clearly V x is C∞, since the Gaussian density is C∞ and all of its derivatives vanish rapidly at( infinity.) To see that V is injective, simply note that if y x, then there exists i 1, 2, 3 such m m m ≠ m ∈ { m } that y x + Ωi , and therefore y + Ωi x + Ωi and hence γ y + Ωi γ x + Ωi . ∈ ⊊ ( ) < ( )

8 To show that V is surjective, fix R and consider the open triangle K x E R = ∈ ∶ max x R . The boundary of K is made up of three line segments, each of the{ form i i < R K x }E x R,x R,x R . By (2.6), we know that for any x K we R,i ∶= ∈ ∶ i = j ≤ k ≤ ∈ R,i have γ x{ Ω 1 Φ R , i.e. that V }x 1 Φ R on K . It follows that for any + i ≤ − i ≤ − R,i v ∆ ( u )∆ min (u ) 1 Φ R , as( )x runs clockwise( ) along ∂K , then V x runs ∈ R ∶= ∈ ∶ i i > − R clockwise{ around v. By Invariance( )} of Domain Theorem [43], V is a homeomorphism( ) between the open triangle K and V K , and since V ∂K ∆ , it follows that R R R ∩ R = ∅ V K must contain ∆ . Since every( v ) int ∆ satisfies( v )∆ for some R , the R R ∈ ∈ R < ∞ surjectivity( ) follows. Finally, we compute at x E: ∈ W x W x n 1 ∇vVi x ∇ve− ( ) dx v, n e− ( ) dH − , ( ) = Sx Ωm = Sx ∂Ωm ⟨ ⟩ + i + i m m where n denotes the outward unit normal to x + ∂Ωi . But note that ∂Ωi j i Σij , = ⋃ ≠ and that the outward unit normal is the constant vector-field ej − ei √2 on Σij . Con- sequently: ( )~ 1 m ∇vVi x Q v,ej − ei Aij x , ( ) = √2 j i⟨ ⟩ ( ) ≠ m and (2.8) is established. Since each of the Aij x is strictly positive, LAm x is non- ( ) singular (in fact, strictly positive-definite) as a quadrat( ) ic form on E, and hence DV x is non-singular as well. It follows that V is indeed a diffeomorphism, concluding( the) proof. We can now give the following: → R Definition 2.4. The model isoperimetric double-bubble profile Im ∶ int ∆ is defined as:

m m m m m Im v ∶ Pγ x + Ω A12 x + A23 x + A31 x such that γ x + Ω v. ( ) = ( ) = ( ) ( ) ( ) ( ) = Lemma 2.3 verifies that the above is well-defined. Thanks to the next lemma, we may (and do) extend Im by continuity to the entire ∆. h Lemma 2.5. Im is C∞ on int ∆, and continuous up to ∂∆. Moreover, if v int ∆ h → 1 ∈ converge to v ∂∆ then Im v Im( ) maxi vi ∶ Im v . ∈ ( ) ( ) = ( ) 1 m 1 ↦ m Proof. Since Im v Pγ V − v + Ω and both V − and the map x Pγ x + Ω are ( ) = ( ( ) ) ( ) C∞ on their respective domains, it follows that Im is C∞ on int ∆. h h 1 h Now suppose that int ∆ v → v ∂∆ and let x V − v . We first consider the case ∋ ∈ = that v is a corner of the simplex, in which case v 1, 0, 0 without( ) loss of generality. Since h m → h h=→( ) h h → γ x + Ω1 1, we see from (2.6) that x1 − x2 −∞ and x1 − x3 −∞. In particular, (h ) h 1 h x , n12 , x , n13 → , and since t23 n12 n13 , it follows that x ,t23 → . It ∞ = √3 + ∞ ⟨ ⟩ ⟨ ⟩ 1 h ( ) h ⟨ ⟩1 follows from (2.5) that γ x Σ → 0 for every i j, and so I v → 0 I( ) 1 . + ij ≠ m = m Finally, consider the case( that v) ∂∆ but is not a corner. Without( ) loss of generality,( ) ∈ h m h m v a, 1 a, 0 for some a 0, 1 . Then γ x Ω1 → a and γ x Ω2 → 1 a; it = − ∈ + + − follows( from (2.6)) that ( ) ( ) ( )

h h h h x1 − x2 1 x2 − x1 1 lim sup Φ− 1 − a , lim sup Φ− a . n→ √ ≤ ( ) n→ √ ≤ ( ) ∞ 2 ∞ 2

9 1 1 Since Φ− 1 − a −Φ− a , and recalling the notation of Lemma 2.1, we conclude that ( ) = ( ) h h h x2 − x1 1 x , n12 → Φ− a . (2.9) ⟨ ⟩ = √2 ( ) Now by (2.7): h 3 h h γ x + Ωm 1 − Φ x , n23 1 − Φ x , n13 , ( ) ≥ ( (⟨ ⟩))( (⟨ ⟩)) n 3 → h h and since γ x( ) + Ωm 0, at least one of x , n23 or x , n13 must diverge to ∞. (h ) ⟨ ⟩ ⟨ ⟩ But since x , n12 converges to a finite limit by (2.9), and since n12 + n23 + n31 0, ⟨ ⟩ h h = it follows that both x , n23 and x , n13 must diverge to ∞. By (2.5), we conclude 1 h 1⟨ h ⟩ ⟨ ⟩ 1 that γ x Σ23 ,γ x Σ13 → 0. In addition, as t12 n31 n32 , it follows + + = √3 + h( ) ( ) ( ) 1 h that x ,t12 → −∞. Together with (2.9), the representation (2.5) implies that γ x + ⟨→ ⟩ 1 1 h → 1 1 ( Σ12 ϕ Φ− a Im( ) a . We thus confirm that Im v Im( ) a Im( ) 1 − a 1 ) ( ( )) = ( ) ( ) ( ) = ( ) = Im( ) maxj vj , as asserted. ( ) 2 We have constructed the model double-bubble clusters in E R for v int ∆ (tripod ≃ ∈ clusters), and on R for v ∂∆ (halfline clusters). Clearly, by taking Cartesian products n 2 n 1 ∈ with R − and R − , respectively, and employing the product structure of the Gaussian measure, these clusters extend to Rn for all n 2. Consequently, I v I v for all ≥ ≤ m v ∆, and our goal in this work is to establish the converse inequality.( ) ( ) ∈

To this end, we observe that Im satisfies a remarkable differential equation: Proposition 2.6. At any point in int ∆:

1 1 2 1 ∇Im V − and ∇ Im − LAm V −1 − = √2 = ( ○ ) as tensors on E. Proof. Let n e e √2 and t e e 2e √6 as in Lemma 2.1. Dif- ij = j − i ij = i + j − k ferentiating (2.5),( and recalling)~ that the one-dimensiona( )~l Gaussian density ϕ satisfies ϕ′ x −xϕ x , we calculate: ( ) = ( ) Am x n x, n Am x t ϕ x, n ϕ x,t ∇ ij = − ij ij ij − ij ij ij ( ) ⟨ ⟩ m( ) (⟨ ⟩) (⟨ ⟩) −nij x, nij Aij x − tij ϕ2 x , = ⟨ ⟩ ( ) ( ) where ϕ2 denotes the standard Gaussian density on E. Since i,j C tij 0, ∑( )∈ =

m m T 1 Aij x Aij x nij nij x LAm x x. ∇ Q = − Q = − ( ) i,j C ( ) i,j C ( ) 2 ( )∈ ( )∈ m 1 Since Im v ∑ i,j C Aij V − v , the chain rule yields ( ) = ( )∈ ( ( )) 1 1 1 1 ∇Im v − LAm V −1 v DV − V − v V − v . ( ) = 2 ( ( ))( ) ( ( )) ( ) 1 1 But according to (2.8), DV − −√2LA− m , and the first claim follows. The second claim ( ) = 1 1 1 follows by differentiating the first one, writing D V − DV − ○V − and applying once 1 1 ( ) = ( ) again DV − −√2LA− m . ( ) =

10 3 Definitions and Technical Preliminaries

n n We will be working in Euclidean space R , ⋅ endowed with a measure γ γ having ( WS S) = C∞-smooth and strictly-positive density e− with respect to Lebesgue measure. We develop here the preliminaries we require for clusters Ω Ω1,..., Ω with k cells, for = k general k 3, as this poses no greater generality over the case( k 3. Recall) that the cells ≥ Rn = k Ωi are assumed to be Borel, pairwise disjoint, and satisfy γ ∖∪i 1Ωi 0. In addition, ( = ) = they are assumed to have finite γ-weighted perimeter Pγ Ωi ∞, to be defined below. ( ) < We denote by T the collection of all cyclically ordered triplets:

T ∶ a,b , b, c , c, a ∶ 1 a b c k . = {{( ) ( ) ( )} ≤ < < ≤ }

We write divX to denote divergence of a smooth vector-field X, and divγ X to denote its weighted divergence: div X divX W. γ ∶= −∇X Given a unit vector n, we write divn⊥ X to denote divX − n, ∇nX , and set its weighted counterpart to be: ⟨ ⟩ divn⊥ X divn⊥ X W. ,γ = −∇X n For a smooth hypersurface Σ