The isope ime ic p oblem in comple e annuli o
e olu ion wi h inc easing Gauss cu a u e
An onio Ca˜ne e
Depa amen o de Ma em´a icas, Uni e sidad de Le´on, 24071 Le´on, Spain
([email p o ec ed])
Manuel Ri o e´
Depa amen o de Geome ´ıa y Topolog´ıa, Facul ad de Ciencias, Uni e sidad de
G anada, 18071 G anada, Spain ([email p o ec ed])
In his wo k we desc ibe he isope ime ic egions in comple e symme ic annuli o e olu ion wi h
Gauss cu a u e non-dec easing om he sho es pa allel. This desc ip ion allows us o comple e
he classifica ion o isope ime ic egions in quad ics o e olu ion.
1. In oduc ion
I is well known ha leas -pe ime e se s o gi en a ea in he plane, in hype bolic
planes and in ound sphe es a e geodesic discs [3]. Howe e , e en in e y simple
su aces, his isope ime ic p oblem has emained open. In 1996, Benjamini and
Cao [2] p o ed ha he leas -pe ime e way o enclose a gi en a ea in a pa aboloid
o e olu ion is by means o a ci cle o e olu ion. This esul was eco e ed by
diffe en me hods by Pansu [9], Topping [12], Mo gan e al. [8] and Ri o ´e [10].
In hese wo ks, he isope ime ic egions we e classified o some new ypes o su -
aces. Benjamini and Cao [2] sol ed he p oblem o comple e planes o e olu ion
wi h non-inc easing cu a u e om he o igin which a e con ex a infini y. Mo gan
e al. [8, §4.3] emo ed his con exi y assump ion, and cha ac e ized he isope i-
me ic egions in eal p ojec i e planes o e olu ion wi h non-inc easing Gauss
cu a u e om he o igin. In [10], amongs o he esul s, he isope ime ic p oblem
was sol ed o sphe es o e olu ion wi h an equa o ial symme y and Gauss cu -
a u e ei he non-inc easing o non-dec easing om he equa o o he poles. An
app oach o he classifica ion o isope ime ic egions in o i o e olu ion has been
gi en by Ca˜ne e [4], who has classified he s able egions in such su aces.
E en in his simple class o examples, he geome y o he pe ime e -minimizing
egions o gi en a ea can be qui e complex. In some planes [8] and sphe es [10] o
e olu ion, hese egions can be ei he discs o annuli, and in annuli o e olu ion
wi h dec easing cu a u e om one end o fini e a ea he isope ime ic egions
a e bounded by a single ci cle o e olu ion [8, 10]. On he o he hand, in o i o
e olu ion, he bounda y o a s able egion can be composed o cu es o cons an
geodesic cu a u e which a e no ci cles o e olu ion [4]. Mo eo e , in non-compac
su aces, isope ime ic egions may no exis . In gene al, a minimizing sequence o
se s o a gi en a ea whose pe ime e s con e ge o he infimum o pe ime e s o his
a ea may ha e a con e gen pa o smalle a ea and a di e ging pa o posi i e
a ea, so ha in he limi we ob ain an isope ime ic egion o smalle a ea, and
possibly some minimizing objec a infini y.
This pape is de o ed o he classifica ion o isope ime ic egions in a com-
ple e annulus o e olu ion wi h an equa o ial symme y and Gauss cu a u e
which is non-dec easing om his equa o . Examples o such annuli a e minimal
ca enoids and one-shee ed hype boloids. We shall use echniques om he calculus
o a ia ions o ea his p oblem by classi ying he embedded cu es wi h con-
s an geodesic cu a u e which can be pa o he bounda y o an isope ime ic
egion. Ou esul s allow us o comple e he classifica ion o isope ime ic egions
in quad ics o e olu ion. In he esolu ion o his p oblem we shall find all he
difficul ies men ioned be o e: non-exis ence o isope ime ic egions, he b eak o a
minimizing sequence in o wo pa s and he exis ence o isope ime ic egions o
se e al diffe en ypes. We p o e in ou main esul , heo em 3.9, ha in a com-
ple e symme ic annulus o e olu ion wi h non-dec easing Gauss cu a u e om
he sho es pa allel, he isope ime ic egions may be
(i) a ‘disc a infini y’,
(ii) a symme ic annulus,
(iii) an asymme ic annulus, o
(i ) an annulus bounded by an unduloid- ype cu e and a ci cle o e olu ion.
All possibili ies occu in diffe en annuli, as shown in example 3.8, whe e we exhibi
isope ime ic egions o he las ype, and in §4.
We ha e o ganized he emainde o he pape in o h ee sec ions. In §2we
es ablish no a ion and gi e p elimina y esul s. In §3, we shall p o e ou main
esul s, mainly ha a minimizing sequence canno be b oken in o wo pieces, and
ha he e exis isope ime ic egions in hese annuli which a e no o e olu ion.
Finally, in §4, we apply he p e ious esul s o he classifica ion o isope ime ic
egions in he one-shee ed hype boloid. This allows us o conclude, in co olla y 4.4,
he classifica ion o quad ics o e olu ion (in a ian by a one-pa ame e g oup o
o a ions a ound a line).
2. P elimina ies
2.1. Annuli o e olu ion wi h non-dec easing cu a u e
We shall deno e by M he p oduc S1×Rendowed wi h a comple e wa ped me ic
ds2:= ( )2dθ2+d 2,
o ∈R,θ∈S1and :R→Ra smoo h posi i e unc ion. The Gauss cu a u e
depends only on he -coo dina e, and is gi en by
K( ):=− ( )
( ).(2.1)
Mo eo e , he leng h and he geodesic cu a u e o he pa allels S1×{ }, a e gi en by
L( ):=2π ( ),h( ):= ( )
( ).(2.2)
We shall suppose ha he annulus Mis symme ic wi h espec o he pa allel
S1×{0}, which is equi alen o he symme y ( )= (− ), and ha he Gauss
cu a u e Kis a non-dec easing unc ion o he dis ance om S1×{0}. We no e
ha he mono onici y o K( ) is equi alen o ha o he unc ion
(4π2)(( )2− )( )=L2(K+h2)( ).
F om he mono onici y o K, since he e a e no comple e ends wi h posi i e cu a-
u e, i ollows ha K⩽0. Mo eo e , i K anishes a some poin 0, hen K≡0
in [ 0,+∞), and so ou annulus is fla nea infini y. Since Kis non-dec easing and
non-posi i e, we shall define
K∞:= lim
→∞ K( )⩽0.
As K⩽0, i ollows ha  ⩾0, and so is non-dec easing. Since (0)=0by
he symme y o , we deduce ha ⩾0 o ⩾0 (and non-posi i e o ⩽0).
Then is non-dec easing o ⩾0 and, consequen ly, Mis comple e and S1×{0}is
a sho es pa allel o M. We will e e o i as he sho es geodesic loop (al hough
i is no necessa ily unique).
Gi en some Ω⊂M, we shall deno e he Riemannian a ea o Mby A(M). I
Cis a ec ifiable cu e, he leng h o Cwill be deno ed by L(C). I Ωis a fini e
pe ime e se in M, hen i s pe ime e will be deno ed by P(Ω).
We shall conside he isope ime ic p oblem o minimizing pe ime e unde an
a ea cons ain in hese symme ic annuli o e olu ion wi h non-dec easing Gauss
cu a u e om he sho es geodesic loop.
2.2. Isope ime ic egions
Fo a su ace M, gi en a∈(0,A(M)), we conside he isope ime ic p ofile o
M, defined by
I(a) = in {L(∂B):B⊂M, smoo h wi h A(B)=a}.
An isope ime ic egion Ω⊂Mis a fini e pe ime e se such ha P(Ω)=I(A(Ω)).
The egula i y esul s by Mo gan [7] imply ha an isope ime ic egion has smoo h
bounda y. The exis ence o an isope ime ic egion o a gi en a ea a>0isno
gua an eed in a non-compac su ace, since a minimizing sequence {Ωn}n∈No se s
o a ea a, and sa is ying
lim
n→∞ L(∂Ωn)=I(a),
may lose all o pa o i s a ea a infini y. Howe e , we ha e he ollowing esul .
Lemma 2.1 (Ri o ´e [10, lemma 1.8]).Le Mbe a Riemannian su ace, A>0, and
le {Ωn}nbe a minimizing sequence o a ea A. Then Ωncan be decomposed as
Ωn = Ωcn ∪ Ωnd, whe e
(i) Ωc
ncon e ges o a se Ω⊂M, wi h A(Ω)∈[0,A],
(ii) Ωd
ndi e ges,
(iii) i Lc= lim L(∂Ωc
n)and Ld= lim L(∂Ωd
n), hen Lc+Ld=I(A), and
(i ) Ωis an isope ime ic egion o a ea A(Ω).
F om his esul , we may conclude ha i he loss o a ea a infini y Ad=
limn→∞ a ea(Ωd
n) is ze o, hen Ωis an isope ime ic egion o a ea A.
2.3. Cons an geodesic cu a u e cu es
Classical a ia ional o mulae o leng h and a ea [11], oge he wi h he eg-
ula i y esul s o isope ime ic egions in su aces, imply ha he bounda y o
an isope ime ic egion has cons an geodesic cu a u e wi h espec o he inne
no mal.
Cons an geodesic cu a u e cu es in o a ionally symme ic su aces can be
classified due o he exis ence o a fi s in eg al coming om he one-pa ame e
g oup o isome ies. F om [4,10] we ha e he ollowing esul .
Theo em 2.2.Le Mbe a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es geodesic loop. Le Cbe a cu e wi h cons an
geodesic cu a u e and fini e leng h.
Then Cis a pa allel, a nodoid- ype cu e o an unduloid- ype cu e.
In nodoid- ype cu es, he angen ec o u ns mono onically, p esen ing poin s
wi h e ical angen ec o . When hese cu es a e closed, hey bound discs in
he su ace. On he o he hand, unduloid- ype cu es a e pe iodic g aphs o e θ,
symme ic wi h espec o e e y c i ical poin o i s -coo dina e. We define he
pe iod o he la e as he θ-dis ance be ween wo consecu i e maxima (o minima)
poin s o he -coo dina e (see figu e 1).
The ollowing lemma gi es necessa y and sufficien condi ions o hese cu es o
be closed and embedded.
Lemma 2.3 (Ri o ´e [10, p oposi ion 1.3]).Le Cbe a cu e wi h cons an geodesic
cu a u e in a wa ped p oduc S1×I.
(i) I Cis a nodoid- ype cu e, i yields a closed embedded cu e i and only i
he maximum and he minimum o |Ca e in he same e ical line.
(ii) I Cis an unduloid- ype cu e, i yields a closed embedded cu e i and only
i he pe iod o Cis equal o 2π/k, wi h k∈N.
2.4. S abili y
We shall say ha a smoo h cu e Co fini e leng h and cons an geodesic cu -
a u e is s able i i is ac ually a local minimum o he pe ime e o a ia ions
θ
Figu e 1. Nodoid- ype and unduloid- ype cu es in he p oduc S1×I.
p ese ing he a ea. Fo such a ia ions, he second de i a i e o leng h is gi en [1]
by
I(u)=−C
ud2u
ds2+(K+h2)uds, (2.3)
whe e u:C→Ris he no mal componen o he ec o field associa ed o he
a ia ion and sis he a c-leng h pa ame e in C. We find ha Cis s able i and
only i
I(u)⩾0 o any unc ion usuch ha C
uds=0.
We will say ha a Ω⊂Mis a s able egion i ∂Ω is an embedded s able cu e
wi h cons an geodesic cu a u e wi h espec o he inne no mal. We ema k ha
any isope ime ic egion is s able.
Rela ed o (2.3), we conside he Jacobi ope a o defined by
J(u)=d2u
ds2+(K+h2)u, (2.4)
and, on e e y connec ed componen C⊂C, he co esponding eigen alue p oblem
J(u)+λu =0,
o C2 unc ions u:C→R. The eigen alues associa ed o he Jacobi ope a o will
be o in e es h oughou his pape . We no e ha a s able cu e canno ha e mo e
han one connec ed componen wi h a nega i e fi s eigen alue. We ecall some
in o ma ion abou hese eigen alues in he ollowing lemma (see [4] o de ails).
Lemma 2.4.Le C⊂Mbe a connec ed cu e wi h cons an geodesic cu a u e, and
conside λ1(C) he fi s eigen alue associa ed o he Jacobi ope a o (2.4) in C.
(i) I Cis a nodoid- ype o an unduloid- ype cu e, hen λ1(C)<0.
(ii) I Cis he pa allel S1×{ }, hen λ1(C)=−(K+h2)( ).
The ollowing esul s discuss he s abili y o he cu es desc ibed in heo em 2.2,
and o he ho izon al annuli bounded by wo pa allels.
Lemma 2.5 (Ri o ´e [10, lemma 1.6]).A pa allel S1×{ }in Mis s able i and only
i
L2(K+h2)( )⩽4π2,(2.5)
o , equi alen ly, i (( )2− )( )⩽1.

Lemma 2.6 (Ri o ´e [10, lemma 1.7]).A ho izon al annulus S1×[ 1,
2]⊂Mis
s able i and only i he pa allels S1×{ i}a e s able (i=1,2) and
L−1(K+h2)( 1)+L−1(K+h2)( 2)⩽0.(2.6)
Mo eo e , in he case o a symme ic annulus S1×[− , ], condi ion (2.6) educes
o
(K+h2)( )⩽0.(2.7)
Lemma 2.7.Le C⊂Mbe a closed embedded nodoid- ype cu e, no con ained in
a egion wi h cons an Gauss cu a u e. Then Cis uns able.
P oo . By [10, lemma 2.3], he only closed embedded nodoids will necessa ily in e -
sec he pa allel S1×{0}. Bu , om [10, lemma 3.4], we conclude ha such nodoids
a e uns able, since S1×{0}is he pa allel whe e Kachie es i s minimum.
Rema k 2.8.We ecall ha closed embedded nodoids con ained in egions wi h
cons an Gauss cu a u e a e s able, by he classical isope ime ic inequali ies.
Applying he same easoning as in [4, §2], i can be p o ed ha he e exis closed
and embedded unduloid- ype cu es (and e en s able ones) in some o ou annuli.
Mo eo e , we shall see in §3 ha hese cu es ac ually appea as he bounda ies
o isope ime ic egions. Le us ecall a necessa y condi ion o he s abili y o
unduloid- ype cu es.
Lemma 2.9 (Ca˜ne e [4, lemma 2.2]).Le C⊂Mbe a closed embedded s able un-
duloid- ype cu e, no con ained in he egion whe e ( )2−  =1. Then he
cu e C ouches he egions whe e ( )2−  <1and ( )2−  >1.
S able symme ic annuli o small a ea sa is y dh/dA>0. These annuli g ow up
o each he pa allels whe e K+h2= 0. A his poin we ob ain s able asymme ic
annuli by le ing one o hese bounda y cu es app oach and he o he mo e away
om he sho es pa allel wi h he same geodesic cu a u e. In his p ocess we
ob ain asymme ic annuli wi h dh/dA<0. The de o ma ion con inues un il he
la ges pa allel in he bounda y o he asymme ic annulus eaches a pa allel wi h
L2(K+h2)=4π2, whe e unduloids appea . This la e phenomenon has been
s udied in de ail in [4].
3. Main esul s
As in [4], om heo em 2.2 we can classi y he s able egions in ou su aces.
Theo em 3.1.Le Mbe a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es pa allel.
Then he s able egions in Mmay be
(i) discs bounded by a nodoid- ype cu e, con ained in a egion wi h cons an
Gauss cu a u e,
(ii) ho izon al annuli symme ic wi h espec o he sho es pa allel, and bounded
by wo pa allels con ained in he egion K+h2⩽0,
(iii) non-symme ic ho izon al annuli bounded by wo pa allels sa is ying he s a-
bili y condi ion (2.6), and con ained in he egion L2(K+h2)⩽4π2,
(i ) annuli bounded by a s able unduloid- ype cu e, and a pa allel con ained in
K+h2<0,o
( ) unions o a disc o cons an Gauss cu a u e, and a symme ic annulus con-
ained in K+h2<0, wi h he same geodesic cu a u e.
P oo . Le Ωbe a s able egion in M. Since he fi s eigen alue o he Jacobi
ope a o (2.4) associa ed o a nodoid- ype o an unduloid- ype cu e is nega i e, i
ollows ha ∂Ω will con ain a mos one o hese cu es. On he o he hand, he
geodesic cu a u e h( ) o pa allels has a diffe en sign in each hal o he annulus
and, consequen ly, ∂Ω will con ain a mos one pa allel in each hal .
I ∂Ω does no con ain ei he a nodoid- ype cu e o an unduloid- ype cu e, hen
Ωis a ho izon al annulus o ype (ii) o (iii) sa is ying he condi ions desc ibed in
lemma 2.6. I ∂Ω has a nodoid- ype cu e, by lemma 2.7 i u ns ou ha Ωis o
ype (i) o ( ). I ∂Ω con ains an unduloid- ype cu e, hen Ωmus be o ype (i ),
sa is ying he condi ion o lemma 2.9.
F om now on we shall deno e by M(K∞) he comple e plane wi h cons an Gauss
cu a u e K∞.
Theo em 3.2.Le Mbe a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es geodesic loop. Conside A>0, and a minimizing
sequence {Ωn}n∈N o a ea A.
I he a ea Aco he con e gen pa and he a ea Ado he di e gen pa a e
bo h posi i e, hen he alue o he isope ime ic p ofile I(A)is gi en by he sum o
he pe ime e o a s able symme ic annulus in Mand he pe ime e o a disc in
M(K∞), bo h wi h he same geodesic cu a u e.
P oo . Since Ldis fini e, he bounda y cu es o he se s o he di e gen pa o he
minimizing sequence a e homo opically i ial o nla ge enough. Since K⩽K∞,
applying he classical isope ime ic inequali y o Ωd
nand passing o he limi we
ge
L2
d⩾4πAd−K∞A2
d.
Conside a disc D⊂M(K∞) o a ea Ad>0. As K∞⩽0, he injec i i y adius
o he complemen o any compac se in Mis infini e. F om he isope ime ic
inequali y i ollows ha
L(∂D)2=4πAd−K∞A2
d⩽L2
d,
whe e Lddeno es he limi leng h o he di e gen pa o he minimizing sequence
{Ωn}n.I L(∂D)<L
d, i is easy o ge a con adic ion om he minimizing cha -
ac e o {Ωn}n, simply by conside ing a amily o geodesic discs in Mo a ea Ad
whose cen es di e ge. Then L(∂D)=Ld.
The abo e easoning shows ha I(A) is gi en by he pe ime e o he union D∪Ω,
whe e Ωis he limi se o he con e gen pa o {Ωn}n∈N. The configu a ion D∪Ω
in M(K∞)∪Mcanno be uns able, since o he wise i could be de o med o a leas
pe ime e configu a ion wi h he same a ea, and he de o ma ion o Dcould be
app oxima ed by a se in M, hus gi ing a con adic ion. Since he fi s eigen alue
associa ed o he Jacobi ope a o (2.4) in Dis nega i e, we conclude ha Ωmus
be a symme ic annulus.
We shall see in he ollowing esul s ha he possibili y o he p e ious heo-
em 3.2 canno hold.
Lemma 3.3.Le Mbe a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es geodesic loop. I Ris a s able symme ic annulus
o a ea A, pe ime e Land geodesic cu a u e h, hen
L > hA.
P oo . We shall p o e he equi alen inequali y L2−LhA > 0. I R=S1×[− , ],
>0, i suffices o check ha
( )2− ( )
0
(s)ds>0.(3.1)
Le gbe he de i a i e wi h espec o o he le -hand e m o (3.1). We ha e
g( )= ( ) ( )+K( ) ( )
0
(s)ds
and
g( )=h( )g( )+m( ),
wi h m( )=K( ) ( )
0 (s)ds.
As g(0) = 0, i ollows (see [6, co olla y 2.1, p. 48]) ha
g( ) = exp 
0
h(s)ds
0
exp −s
0
h(u)dum(s)ds⩾0.
The e o e, he le -hand e m in (3.1) is an inc easing unc ion, and he desi ed
inequali y holds.
P oposi ion 3.4.Le Mbe a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es geodesic loop. Le h>0. Then he union o a
s able symme ic annulus Rhand a disc Dhin M(K∞)wi h he same geodesic
cu a u e hhas a la ge pe ime e han he disc Din M(K∞)o a ea A(Rh)+
A(Dh).
P oo . Le −b2:= K∞and A:= A(Rh)+A(Dh). F om he isope ime ic inequali y
in M(−b2)weha e
L(∂D)2=4πA +b2A2.
On he o he hand, i is easy o check ha , o he geodesic disc Dho geodesic
cu a u e hin M(−b2), we ha e
hL(∂Dh)=2π+b2A(Dh),(3.2)
h>b. (3.3)
Le us p o e ha
(L(∂Dh)+L(∂Rh))2>4πA +b2A2.(3.4)
By applying he isope ime ic inequali y in M(−b2) oDh, and lemma 3.3 o Rh,
we ha e
(L(∂Dh)+L(∂Rh))2=L(∂Dh)2+L(∂Rh)2+2L(∂Dh)L(∂Rh)
>4πA(Dh)+b2A(Dh)2+h2A(Rh)2+2hL(∂Dh)A(Rh).
(3.5)
F om (3.3) i is clea ha
h2A(Rh)2>b
2A(Rh)2,(3.6)
Finally, by using (3.2) and (3.6), we ob ain om (3.5) ha
(L(∂Dh)+L(∂Rh))2>4π(A(Dh)+A(Rh)) + b2(A(Dh)+A(Rh))2,
which p o es he s a emen .
F om p oposi ion 3.4 we ob ain wo in e es ing consequences.
Co olla y 3.5.Conside a minimizing sequence o a ea A. Then he a eas Ad,
Aco he di e gen and con e gen pa s canno be posi i e simul aneously.
Co olla y 3.6.The alue o he isope ime ic p ofile o some gi en a ea A>0
canno be achie ed by he sum o he pe ime e s o a s able symme ic annulus and
a disc in M(K∞).
In o he wo ds, he union o a s able symme ic annulus and a disc in M(K∞)
wi h he same geodesic cu a u e ‘canno be an isope ime ic egion’ in M.
Example 3.7.Le Mbe a minimal ca enoid defined by
x2+y2=λ2cosh2z
λ,λ>0.
This su ace is included in ou amily o annuli, and he co esponding wa ped
unc ion is
( )=( 2+λ2)1/2, ∈R.
As ( )2−  <1, he e a e no s able closed unduloid- ype cu es in M.Bya
compa ison a gumen , i ollows ha he discs in M(K∞)=M(0) ha e a smalle
pe ime e han ho izon al annuli, and so he isope ime ic p ofile o he ca enoids
is gi en by he plana isope ime ic inequali y I(a)=(4πa)1/2. The alidi y o
he plana isope ime ic inequali y in minimal su aces is an ex emely in e es ing
subjec (see [5]).
We now gi e an example o a symme ic annulus o e olu ion wi h non-dec easing
Gauss cu a u e om he sho es pa allel, whe e he la ges a ea s able asymme ic
annulus has a smalle pe ime e han a disc in M(K∞). Then, by he con inui y
o he isope ime ic p ofile, i u ns ou ha annuli bounded by an unduloid- ype
cu e and a pa allel a e also isope ime ic egions.