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. The conjec u al con igu a ion o n= 6
As be o e, we beli e ha he con igu a ions o Figu e 8 below a e s able, bu es ima es
done wi h he Su ace E ol e conside ing equal a eas show ha hey a e nonminimizing.
18 A. CA˜
NETE AND M. RITOR´
E
(1) (2) (3)
Figu e 8. Some o he con igu a ions o n= 6
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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