Least-perimeter Partitions of the Disk into Three Regions of Given Areas

LEAST-PERIMETER PARTITIONS OF THE DISK INTO THREE REGIONS OF GIVEN AREAS ANTONIO CA˜ NETE AND MANUEL RITOR´ E Abs ac . We p o e ha he unique leas -pe ime e way o pa i ioning he uni 2-dimen- sional disk in o h ee egions o p esc ibed a eas is by means o he s anda d g aph desc ibed in Figu e 1. In oduc ion Pa i ioning p oblems in he Calculus o Va ia ions ha e mul iple applica ions in physical sciences. They can model mul i ude o na u al phenomena such as he shape o a cellula issue, he in e ace o sepa a ion be ween luids, and many o he s, as desc ibed in he ea ise by D’A cy Thompson [Th]. In his wo k we conside he isope ime ic p oblem o pa i ioning a plana disk in o h ee egions o gi en a eas wi h he leas possible pe ime e , and we p o e ha he s anda d con igu a ion in Figu e 1, consis ing o h ee ci cula a cs o segmen s mee ing o hogonally he bounda y o he disk, and mee ing in h ees a 120 deg ees in an in e io e ex, is he only solu ion o his p oblem. Figu e 1. The leas -pe ime e pa i ion o he disk in o h ee gi en a eas In addi ion o he abo e condi ions, he solu ion mus sa is y a ce ain balancing condi ion on he geodesic cu a u es o he ci cles. This condi ion will be s a ed p ecisely in he nex sec ion. Exis ence and egula i y o solu ions o his p oblem a e gua an eed by he esul s o F. Mo gan [M2], who showed ha he minimize , in he in e io o he disk, is composed o smoo h cu es o cons an geodesic cu a u e mee ing in h ees a 120 deg ee angles. Da e: June 25, 2003. 2000 Ma hema ics Subjec Classi ica ion. 49Q10, 51M25, 52A38, 52A40. Key wo ds and ph ases. Isope ime ic pa i ion, s abili y, s able. Bo h au ho s ha e been suppo ed by MCyT-Fede esea ch p ojec BFM2001-3489. 2 A. CA˜ NETE AND M. RITOR´ E Bounda y egula i y also ollows om [M2] al hough i is no explici ly s a ed in his wo k. Exis ence and egula i y in highe dimension we e s udied by F. Almg en [Alm]. The leas -pe ime e way o pa i ioning a disk Din o wo egions o gi en a eas is by means o an a c o ci cle o segmen ha mee s o hogonally ∂D. F om he exis ence and egula i y esul s in nex sec ion i ollows ha he e is a solu ion, which is a smoo h, possibly nonconnec ed, embedded cu e wi h cons an geodesic cu a u e ha mee s ∂D o hogonally. Such a cu e mus be connec ed, since o he wise we could o a e one componen wi h espec o he cen e o he disk un il i ouches a second one, hus p oducing a non-allowed singula i y. On he o he hand, as he cu e has cons an geodesic cu a u e, i mus be pa o a ci cle o o a line. Figu e 2. The leas -pe ime e pa i ion o he disk in o wo gi en a eas The isope ime ic p oblem consis ing o enclosing ngi en a eas in he disk o in he plane wi h he leas possible pe ime e has a s ong complexi y which is de i ed no om he geome y o he indi idual componen s o he solu ion ( hey can be desc ibed in e ms o ci cles o lines) bu om hei la ge numbe . The plana double bubble conjec u e was p o ed by J. Foisy e al. [FABH], who showed in 1993 ha he s anda d plana double bubble uniquely minimizes pe ime e in R2. Assuming ha he s udied egions a e connec ed, C. Cox e al. [CHH] p o ed in 1994 ha he s anda d plana iple bubble uniquely minimizes pe ime e in he plane o any h ee gi en a eas. R. P. De e eaux [D] s udied in 1998 he plana iple bubble conjec u e unde he hypo hesis ha all he egions ha e he same p essu e. W. Wichi amala inally p o ed he plana iple bubble conjec u e in 2002 in his Ph. D. Thesis [W]. J. Mas e s [M] p o ed in 1996 he double bubble conjec u e in S2. In e es ing p elimina y wo k was ca ied ou by Bleiche [B1], [B2], [B3]. Conce ning bounda y p oblems, G. H uska e al. [HLPS] ha e ob ained some esul s o plana bubbles in co ne s. Also esul s on o i and cones ha e been ob ained in [CHLL], [BL]. In highe dimensions, J. Hass and R. Schla ly [HS] p o ed he double bubble conjec u e in R3 o equal olumes. The gene al conjec u e was se led by M. Hu chings e al. [HMRR]. Fo highe dimensional Euclidean spaces Reicha d e al. [RHLS] ha e ob ained a p oo o he double bubble conjec u e in R4and pa ial esul s in highe dimensional Euclidean spaces. In he h ee-dimensional o us, M. Ca ion e al. [CCWB] ha e p o ided nume ical e idence o a double bubble conjec u e wi h en ypes o solu ions. In he h ee-dimensional sphe e and he h ee-dimensional hype bolic space, A. Co on and D. F eeman [CF] ha e also ob ained pa ial esul s on he conjec u e ha he s anda d double bubble in hese spaces uniquely minimizes pe ime e . LEAST-PERIMETER PARTITIONS OF THE DISK 3 Plana bubbles a e also o g ea in e es o physicis s. In e es ing a icles ocusing on physical aspec s o he p oblem a e [GJJF] and [CGFM]. The mos in e es ing ma hema ical open ques ion o hese p oblems is o show ha he minimizing con igu a ions mus ha e connec ed egions, ei he in R2o in he disk. In ad- di ion, in he plana p oblem, one should also be able o p o e ha he ex e io egion is connec ed, i.e., ha he e a e no emp y chambe s. We ha e o ganized his pape in se e al sec ions. In Sec ion 1 we gi e p ecise de ini ions, compu e he i s and second a ia ions o leng h o g aphs, ecall exis ence and egula i y esul s o he p oblem o minimizing pe ime e while pa i ioning he disk in o gi en a eas, and s a e some p ope ies which minimizing g aphs mus sa is y. In Sec ion 2 we ob ain a bound on he numbe o componen s o he la ges p essu e egion de e mined by a g aph which minimizes pe ime e up o second o de . We conclude ha a minimizing con igu a ion mus ha e one o en possible ypes, desc ibed in Figu e 4. In Sec ion 3 we p o e he necessa y esul s o disca d he possibili ies ob ained in Sec ion 2, which allow us o p o e ou Main Theo em in Sec ion 4. In a inal sec ion, we indica e u he lines o esea ch and gi e se e al conjec u es. All he pic u es in his pape ha e been made by using Su ace E ol e , a so wa e de eloped by Ken B akke (h p://www.susqu.edu/ acs a /b/b akke/). 1. P elimina ies 1.1. No a ion. Le D⊂R2be he closed uni disk in cen e ed a he o igin. An admissible g aph C⊂Dconsis s o e ices and cu es so ha a e e y in e io e ex ( ha is, a e ex in he in e io o D) h ee cu es o Cmee and a e e y bounda y e ex (a e ex in ∂D) jus one cu e o Cmee s ∂D. We shall also assume ha Cinduces a decomposi ion o he open uni disk in o n egions Ri, 1 ⩽i⩽n, possibly nonconnec ed. An m-componen is a connec ed componen o a egion wi h medges. I Riand Rja e adjacen egions, we will deno e by Cij ⊂C he (no necessa ily connec ed) cu e sepa a ing hem. Le I(i) = {j6=i;Rj ouches Ri}. Wi h his no a ion ∂Ri∩in (D) = [ j∈I(i) Cij. We shall deno e by Nij he no mal ec o o he cu e Cij poin ing in o he egion Ri, and by hij he geodesic cu a u e o he cu e Cij wi h espec o he no mal Nij. As anda d g aph consis s in h ee ci cula a cs o lines segmen s mee ing a an in e io e ex a 120 deg ee angles, eaching o hogonally ∂D, and so ha he sum o he geodesic cu a u es is ze o. Gi en nposi i e numbe s a1,...,ansuch ha Pn i=1 ai=π, he isope ime ic p o ile is he unc ion I(a1,...,an) de ined as he in imum o he leng hs o all admissible g aphs sepa a ing egions in he disk o a eas a1,...,an. We will say ha an admissible g aph Cis minimizing o p esc ibed a eas a1,...,ani I(a1,...,an) is a ained by C. 4 A. CA˜ NETE AND M. RITOR´ E 1.2. Va ia ional o mulae. Gi en an admissible g aph C⊂D, we will conside smoo h one-pa ame e a ia ions ϕ :C→D o small, which sa is y ϕ (∂D)⊂∂D. We will deno e by X=dϕ /d | =0 he associa ed in ini esimal ec o ield, which is smoo h on e e y cu e Cij. No e ha X(p) is angen o ∂D o each pin ∂D. Le uij =X·Nij be he no mal componen o Xon Cij. Gi en such a a ia ion, i is easy o check ha he de i a i e o he a ea Aio Ria = 0 is gi en by (1.1) dAi d  =0 =−X j∈I(i)ZCij uij. Fo he de i a i e o leng h o such a a ia ion we ha e P oposi ion 1.1 (Fi s a ia ion o leng h [HMRR, Lemma 3.1]).Conside an admissible g aph C⊂D, and a smoo h a ia ion ϕ :C→Dwi h associa ed ec o ield X. Then he i s de i a i e o he leng h o ϕ (C)a = 0 is gi en by (1.2) dL d  =0 =−1 2X i∈{1,...,n} j∈I(i) ZCij hijuij +X p∈∂Cij X(p)·νij(p), whe e νij(p)is he inne cono mal o Cij in p. We will say ha an admissible g aph is s a iona y i (1.2) anishes o any a ea-p ese ing a ia ion. F om P oposi ion 1.1 i is easy o p o e he ollowing P oposi ion 1.2. Le C⊂Dbe a s a iona y g aph. Then he ollowing condi ions a e sa is ied (i) The geodesic cu a u e hij is cons an on Cij. (ii) The edges o Cmee in h ees a 120-deg ee angles in in e io e ices. (iii) The balancing condi ion: h ee edges Cij ,Cjk,Cki mee ing in an in e io e ex sa is y (1.3) hij +hjk +hki = 0. (i ) The edges o Cmee ∂D o hogonally a bounda y e ices. Condi ion (ii) implies ha , in some in e io e ex whe e he h ee cu es Cij,Cjk,Cki mee , he no mals add up o ze o, i. e., Nij +Njk +Nki = 0. This implies ha he no mal componen s o he ec o ield Xmus sa is y (1.4) uij +ujk +uki = 0. Gi en a s a iona y g aph C, and a unc ion u:Si,j Cij →R, wi h uij =u|Cij , sa is ying condi ion (1.4) on e e y in e io e ex, i is always possible o ind a ec o ield Xon C, so ha uij =X·Nij and Xis angen o ∂D in each bounda y e ex. Associa ed o Xone can also ind a one-pa ame e a ia ion ϕ :C→D, o small enough, so ha ϕ (p) = expp( X(p)) o any pou o an a bi a ily small neighbou hood o ∂D. The a gumen is as ollows: ix an a bi a y neighbou hood Uo ∂D ha does no con ain in e io e ices o C. Modi y Xso ha i is no mal o Cin U. Le νbe he inne no mal o ∂D. Ex end i o Uso ha i is angen o he edges o C. Also ex end X o a ec o ield on U∩D by means o he exponen ial mapping. Le λbe a smoo h unc ion equal o 1 nea ∂D wi h LEAST-PERIMETER PARTITIONS OF THE DISK 5 suppo in U. Conside he ec o ield Y=X−(X·(λν)) λν and he local one-pa ame e g oup ψ gene a ed by Y. Since νis angen o Cand Xis no mal o Cin U, we ha e ha Y=Xon C. Mo eo e , o p∈∂D, he ec o Y(p) is angen o ∂D. Hence he de o ma ion ψ (C∩U) has ini ial eloci y ec o ield Xand keeps Cinside he disk. The a ia ion ψ (C) has he u he p ope y ha coincides wi h expp( X(p)) in Uou o he suppo o λ. Now we simply de ine ϕ (p) equal o expp( X(p)) ou o he suppo o λ, and equal o ψ (p) in U. The balancing condi ion (1.3) allows us o de ine a p essu e pion e e y egion Ri, s a ing om a gi en egion, so ha (1.5) hij =pi−pj. These p essu es a e de e mined up o an addi i e cons an . The i s a ia ion o mula o leng h can be ew i en in e ms o p essu es in he ollowing way: i Cis a s a iona y g aph, hen he i s a ia ion o leng h o an a bi a y a ia ion is gi en by: (1.6) dL d = n X i=1 pi dAi d . Obse e ha he inde e mina ion o he p essu es up o some addi i e cons an does no a ec he abo e o mula since Pn i=1 dAi/d = 0 o any a ia ion o he egions Ri, as Pn i=1 Ai( ) = a ea(D) along he a ia ion. Le us p o e now he second a ia ion o mula o leng h P oposi ion 1.3 (Second a ia ion o leng h).Le Cbe a s a iona y g aph and le {ϕ }be a a ia ion wi h associa ed ec o ield Xp ese ing a eas up o second o de . Then he second de i a i e o leng h a = 0 is gi en by −1 2X i=1,...,n j∈I(i) ZCij (u′′ ij +h2 ijuij)uij +X p∈∂Cij p∈in (D) −qiju2 ij +uij ∂uij ∂νij (p)(1.7) +X p∈∂Cij p∈∂D u2 ij +uij ∂uij ∂νij (p), whe e qij(p) = (hki +hkj)(p)/√3, and Rkis he hi d egion ouching he e ex p. P oo . Di e en ia ing he in eg al e ms in equa ion (1.2), we ge d d  =0ZCij hijuij=ZCij (u′′ ij +h2 ijuij)uij +hij d d  =0ZCij uij, bu since he a ia ion p ese es a eas up o second o de , i ollows ha X i∈{1,...,n} j∈I(i) hij d d  =0ZCij uij= 2 n X i=1 pi d2 Ai d 2 =0 = 0. Di e en ia ing now he second e m in equa ion (1.2), we ge d d  =0 (X·νij) = (DXX·νij) + uij hij (X·νij) + uij ∂uij ∂νij . 6 A. CA˜ NETE AND M. RITOR´ E Fo p∈in (D), he i s e m anishes since νij +νjk +νki = 0, and a e some calcula ions as in [HMRR], he second one can be seen as −qiju2 ij, whe e qij = (hki +hkj)/√3. Fo p∈∂D, since he con igu a ion is s a iona y, he edges mee ∂D o hogonally, so ha DXX(p)·νij (p) equals u2 ij imes he geodesic cu a u e o ∂D, and (X·νij )(p) = 0.  The condi ion ha he a ia ion mus p ese e a ea up o second o de is no eally needed as we can show in he nex Lemma Lemma 1.4. Le C⊂Dbe a s a iona y g aph. Gi en smoo h unc ions uij :Cij →Rsuch ha (1.1) and (1.4) a e sa is ied (a a ia ion ha p ese es a ea up o i s o de is gi en), he e is a a ia ion {ϕ }o Cwhich lea es cons an he a ea o he egions enclosed by ϕ (C) and such ha he no mal componen s o he ini ial eloci y ec o ield Xa e he unc ions uij. P oo . Le Xbe a ec o ield on C, smoo h o e each cu e Cij, such ha X·Nij =uij. Le ψ :C→Dbe a one-pa ame e a ia ion o Cassocia ed o Xsuch ha ψ (p) = expp( X(p)) ou o a small neighbou hood Uo ∂D which does no con ain in e io e ices o C. We label he egions Riso ha Ri ouches Ri+1 o i= 1, ...,n−1. Choose posi i e unc ions iwi h suppo in he in e io o Ci(i+1) and ou o U. The a ia ion induced by he ec o ield iNi(i+1) dec eases he a ea o Ri, inc eases he a ea o Ri+1 and lea es cons an he a ea o he emaining egions. Conside he a ia ion equal o ( , s1,...,sn−1)7−→ expp X(p) + n−1 X i=1 si iNi(i+1)(p),in C∩(D−U), and equal o ψ (p) o p∈C∩U. Conside he unc ion (A1,...,An−1) o ( , s1,...,sn−1), gi en by he a eas o he de o ma ion o he egions R1,...,Rn−1. The Jacobian ∂(A1,...,An−1) ∂(s1,...,sn−1) is lowe iangula , wi h non- anishing en ies in he p incipal diagonal, so ha he ma ix is egula . The Implici Func ion Theo em allows us o ind smoo h unc ions s1( ),...,sn−1( ) such ha Ai( , s1( ),...,sn−1( )) is cons an o all i. The ini ial eloci y ec o ield o such a a ia ion is equal o Xon C∩U, and o X+ Pn−1 i=1 s′ i(0) iNi(i+1) on C∩(D−U). As s′ i(0) = 0 since ψ p ese es a eas up o i s o de , we conclude ha Xis he ini ial eloci y ec o ield.  Rema k 1.5.A a ia ion o a s a iona y g aph Cby s a iona y g aphs p ese es he angles be ween edges a in e io e ices and he o hogonali y condi ion a bounda y e ices. Gi en a a ia ion p ese ing he a ea o all he egions up o i s o de , we can modi y i by Lemma 1.4 so ha he a eas enclosed a e cons an along he de o ma ion. F om he second a ia ion o mula we ge ha he second de i a i e o leng h is gi en by d2 L d 2=X α dpα d dAα d , whe e αlabels he componen s o he s a iona y g aph ( egions can be disconnec ed), and dpα/d is he de i a i e o he p essu e o he componen αwi h espec o he conside ed a ia ion. Take in o accoun ha he quan i y u′′ ij +h2 ijuij, he de i a i e o he geodesic LEAST-PERIMETER PARTITIONS OF THE DISK 7 cu a u e hij, only depends on uij , he no mal componen o he a ia ional ec o ield X, and ha he modi ica ion needed in Lemma 1.4 o p ese e a eas only modi ies he accele a ion o he a ia ion. The angle-p ese ing condi ion depends only on he ini ial eloci y ec o ield. In gene al, i he a eas a e no p ese ed up o second o de , he second de i a i e o leng h, o a de o ma ion o a s a iona y g aph by s a iona y g aphs, is gi en by d2 L d 2=X α dpα d dAα d +pα d2 Aα d 2, which can also be ob ained by di e en ia ing equa ion (1.6). Rema k 1.6.Fo a a ia ion such ha he angles be ween he edges a e p ese ed, we ha e DX(νij +νjk +νki) = 0 (since νij +νjk +νki = 0 o all ), so he bounda y e m in he second a ia ion o mula anishes. 1.3. Admissible unc ions and he index o m. Le Cbe a s a iona y g aph. We say a unc ion u:Si,j Cij →Ris admissible i he es ic ions uij =u|Cij lie in he Sobole space W1,2(Cij), and e i y ha a any in e io e ex p,uij(p) + ujk(p) + uki(p) = 0. These unc ions co espond o a ia ions o Cwhich ha e as no mal componen s o he associa ed ec o ield X he unc ions uij . These a ia ions will p ese e a eas i , o each i, X j∈I(i)ZCij uij = 0. An admissible unc ion uis said o be a Jacobi unc ion i he associa ed a ia ion p ese es he geodesic cu a u es o each edge Cij and he angles in each e ex. The ac ha he geodesic cu a u es a e p ese ed means ha he es ic ions uij o Cij e i y u′′ ij +h2 ijuij = 0. I is clea ha he no mal componen o he Killing ec o ield gene a ed by he o a ions abou he o igin gi es a Jacobi unc ion. F om equa ion (1.7), we de ine he index o m, ha is, he induced bilinea o m de ined on he space o admissible unc ions, by Q(u, ) = −1 2X i=1,...,n j∈I(i) ZCij (u′′ ij +h2 ijuij) ij (1.8) +X p∈∂Cij p∈in (D) −qijuij +∂uij ∂νij (p) ij(p) + X p∈∂Cij p∈∂D uij +∂uij ∂νij (p) ij(p), whe e qij a e he unc ions de ined in P oposi ion 1.3. We will say a s a iona y g aph Cis s able i Q(u, u)⩾0 o any admissible unc ion u whose associa ed a ia ion p ese es a eas, and uns able i i is no s able. I is clea ha a minimizing con igu a ion mus be s able. 8 A. CA˜ NETE AND M. RITOR´ E 1.4. Exis ence and Regula i y. F om he esul s o F. Mo gan [M2], we ob ain he o- llowing Theo em 1.7 (Exis ence and Regula i y [M2, Th. 2.3]).Le D⊂R2be a closed disk, and le a1,...,anbe ngi en a eas such ha Pn i=1 ai= a ea(D). Then he e exis s a g aph sepa a ing Din o n egions o a eas a1,...,an. Mo eo e such a g aph consis s o cons an geodesic cu a u e cu es mee ing in h ees in he in e io o Da 120 deg ee angles, sa is ying he balancing condi ion (1.3) o he geodesic cu a u es, and mee ing ∂D, one a a ime, in an o hogonal way. P oo . F om he esul s in [M2] one ge s he exis ence o a solu ion and he egula i y in he in e io o he disk wi h jus iple poin s as possible singula i ies. One also ge s ha he e is a ini e numbe o componen s (and hence o cu es) in he minimizing con igu a ion. Fo he bounda y egula i y, we only need o p o e ha a e e y poin o ∂D, a mos one cu e o he minimizing con igu a ion a i es, a 90 deg ees. I one o se e al cu es mee ∂D a pand a leas one o hem is no o hogonal o ∂D, hen he i s a ia ion o mula implies ha he g aph is no s a iona y. Suppose now ha se e al cu es mee o hogonally ∂D a p. We o de hem coun e -clockwise and we conside he i s one, C, which is he common bounda y o componen s Ωiand Ωj, wi h Ωia bounda y one. Make a small de o ma ion in he in e io o Cwhich implies a loss o a ea δ o Ωi. In o de o p ese e he a eas, i is possible o choose a poin qnea pin C, join q o he second cu e C′, which is in he bounda y o Ωj, and elimina e he pa o Cbe ween pand q. I can be checked ha his new con igu a ion, o δsmall enough, educes pe ime e . Then we ge he desi ed egula i y in he bounda y o D. 1.5. Some p ope ies o minimizing g aphs. We now gi e and ecall some esul s on minimizing g aphs ha will be used o p o e ou main heo em Lemma 1.8. Gi en nposi i e numbe s a1,...,ansuch ha Pn i=1 ai=π, we ha e (1.9) I(a1,...,an)⩽n. Mo eo e , equali y is ne e achie ed o n⩾4. I equali y holds in he case n= 3 hen he s anda d g aph consis ing o h ee line segmen s di iding he disk in o h ee egions o equal a eas is minimizing. P oo . We can di ide he disk in o egions o gi en a eas a1,...,anby using app op ia e n adii. This gi es (1.9). Fo n⩾4, his con igu a ion has a p ohibi ed singula i y a he o igin, so ha i canno be minimizing. I equali y holds in (1.9) o n= 3, he con igu a ion mus be s a iona y, so ha he h ee adii mee in 120 deg ees, and he con igu a ion is he s anda d one o equal a eas.  Lemma 1.9. A minimizing g aph mus be connec ed. P oo . On a nonconnec ed g aph, we can o a e one o he componen s un il i ouches ano he one c ea ing an i egula mee ing, so he g aph canno be minimizing.  Rema k 1.10.Le C⊂Dbe a minimizing g aph, and Ω a connec ed componen o D−C. Lemma 1.9 implies ha ∂Ω∩∂D has o be connec ed. Lemma 1.11 ([FABH, Lemma 2.4]).On a minimizing g aph, he e a e no 2-componen s. LEAST-PERIMETER PARTITIONS OF THE DISK 9 2. A bound on he numbe o componen s o he la ges p essu e egion Lemma 2.1. Le Cbe a s able g aph sepa a ing Din o n egions. Then he egion o la ges p essu e has a mos n−1nonhexagonal componen s. P oo . Assume R1is he egion o la ges p essu e and suppose i has a leas nnonhexagonal componen s, Ω1,...,Ωn. Fo each i, conside he a ia ion gi en by ui= 1 on ∂Ωi, ex ended by ze o o he whole g aph. I Ωiis a bounda y componen hen Q(ui, ui) = −X j∈I(1) ZC1j∩∂Ωi h2 1j+X p∈∂C1j∩∂Ωi p∈in (D) −q1j(p) + X p∈∂C1j∩∂Ωi p∈∂D 1<0, since, o pin C1j∩C1k∩∂Ωi, we ha e q1j(p) + q1k(p) = hk1+hkj +hj1+hjk √3(p) = hk1+hj1 √3(p)⩽0, as R1has he la ges p essu e. I Ωiis an in e io componen hen Q(ui, ui) can be compu ed as abo e excep ha he las summand does no appea . So we ge Q(ui, ui)⩽0, and equali y holds i and only i Ωiis bounded by segmen s. I is easy o ob ain, om Gauss- Bonne Theo em, ha Ωihas o be an hexagon. In he case o h ee egions, his only occu s i he h ee p essu es a e equal. Hence, in ou case we can ind some non i ial linea combina ion uo ui, such ha he induced a ia ion p ese es a eas up o i s o de and Q(u, u)<0.  Lemma 2.2. Le C⊂Dbe a minimizing g aph sepa a ing Din o h ee egions. Then Cis one o he g aphs in Figu e 4. P oo . Suppose i s ha all he p essu es a e equal. I all he componen s ouch he bounda y o D hen Cis s anda d. I he e is an in e io componen , hen i is hexagonal. I is easy o see ha he edges lea ing he e ices o he hexagon mee ∂D (o he wise we could ind wo di e en pa allel ays mee ing o hogonally ∂D). This implies ha he g aph is like in Figu e 3. This g aph has wo egions wi h h ee nonhexagonal con ex componen s, and so i is uns able by Lemma 2.1. 2 3 2 13 32 Figu e 3. A g aph wi h an hexagonal in e io componen 16 A. CA˜ NETE AND M. RITOR´ E We now elimina e con igu a ions (4) and (5). These con igu a ions p esen an in e io 4-componen o R1, wi h h ee inciden edges mee ing he ex e io o he disk. I we ex end he ou h edge, i will mee ∂D o hogonally by P oposi ion 3.3, and we will ob ain a con ig- u a ion o ype (3). Hence, he in e io 4-componen has wo o hogonal symme ies mee ing a he o igin, and we conclude as be o e he exis ence o ou nodal egions. Conside now con igu a ion (6). Fix an in e io 4-componen Ω o R1. I we ex end he edge lea ing he bounda y o Ω ha does no each ∂D, i will mee ∂D o hogonally due o he exis ence o a symme y o Ω which is in ac a symme y o he disk. In his way we ob ain a con igu a ion o ype (3). As abo e, Ω will ha e wo o hogonal symme ies mee ing a 0 and so we can ge ou nodal egions yielding ins abili y. Conside now con igu a ion (7). Applying Lemma 3.9, he wo 4-componen s will be symme ic abou wo lines 1, 2passing h ough he cen e o he disk ( he co esponding edges a e no coci cula ). Le q1, q2be he in e sec ion poin s o each line wi h he in e io edges o hese componen s, ha will be ze os o he Jacobi ield uinduced by he one- pa ame e g oup o o a ions abou he o igin. The e lec ion o q1wi h espec o 2lies in he bounda y o he 3-componen o R1and i is no a e ex o he con igu a ion. This poin is clea ly also a ze o o u. Then uhas ou nodal egions and he con igu a ion is uns able. By P oposi ion 3.11, con igu a ion (8) is no minimizing. Con igu a ion (9) is uns able: i he op and bo om edges o each componen a e coci cula hen he con igu a ion is no minimizing by Lemma 3.8. O he wise we can ind an ho izon al symme y o he g aph, which is also a symme y o he disk by Lemma 3.9. Each in e io 4- componen has a e ical symme y so ha he in e io componen s o R1a e iden ical. Using he unc ion equal o +1 on one o hese componen s, equal o −1 on he o he componen , and ze o o he wise, we ha e ob ained a unc ion sa is ying he mean alue condi ions (1.1) which is nega i e o he index o m. So his con igu a ion is uns able. We could also use he me hod o P oposi ion 3.11 o see ha his con igu a ion is nonminimizing. So he only emaining possibili y is con igu a ion (10), he s anda d one. Uniqueness o gi en a eas comes om Theo em 3.6.  5. Final ema ks In his pape we ha e ob ained ha he p oblem o di iding he disk in o h ee a eas has a unique solu ion in which all egions a e connec ed, as in he p oblem o pa i ioning he disk in o wo a eas. I is na u al o conjec u e ha Conjec u e 5.1.A minimizing g aph sepa a es he disk in o connec ed egions. I we conside he p oblem o n egions, wi h n⩾4, by Lemma 2.1 he egion o la ges p essu e will ha e a mos n−1 nonhexagonal connec ed componen s and we can ob ain by combina o ial a gumen s a lis o all possible minimizing con igu a ions. O cou se he numbe o candida es inc eases e y apidly when he numbe o egions inc eases. We belie e ha he ollowing conjec u es a e ue Conjec u e 5.2.The leas pe ime e way o di iding he uni disk in o ou egions o p es- c ibed a eas is gi en by con igu a ion (1) o Figu e 6. LEAST-PERIMETER PARTITIONS OF THE DISK 17 Conjec u e 5.3.The leas pe ime e way o di iding he uni disk in o i e egions o p esc ibed a eas is gi en by con igu a ion (2) o Figu e 6. (1) (2) Figu e 6. The conjec u al con igu a ions o n= 4 and n= 5 Fo each case, we belie e ha he e is ano he possibly s able con igu a ion: o n= 4, he con igu a ion wi h h ee bounda y egions su ounding an in e io one o h ee edges, and o n= 5, he one consis ing in ou bounda y egions su ounding an in e io egion o ou edges. Bu es ima es we ha e done using Su ace E ol e (Ken B akke, 1992) o equal a eas show ha hey a e nonminimizing. Fu he mo e, o n= 4, i any o he a eas ends o ze o, we should ob ain in he limi he s anda d con igu a ion o h ee a eas, which also disca ds he con igu a ion desc ibed abo e a leas o some a eas. In he case n= 5 we should ha e he same beha iou . Fo n= 6 we gi e he ollowing conjec u e Conjec u e 5.4.The leas pe ime e way o di iding he uni disk in o six egions o p esc ibed a eas is gi en by con igu a ion o Figu e 7. Figu e 7. 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Depa amen o de Geome ´ ıa y Topolog´ ıa, Facul ad de Ciencias, Uni e sidad de G anada, E-18071 G anada (Espa˜ na) E-mail add ess:an oni[email p o ec ed] Depa amen o de Geome ´ ıa y Topolog´ ıa, Facul ad de Ciencias, Uni e sidad de G anada, E-18071 G anada (Espa˜ na) E-mail add ess: i o e@ug .es