Free boundary CMC annuli in spherical and hyperbolic balls

Calc. Va . (2025) 64:37
h ps://doi.o g/10.1007/s00526-024-02883-6
Calculus o Va ia ions
F ee bounda y CMC annuli in sphe ical and hype bolic balls
Albe o Ce ezo1,2 ·Isabel Fe nández2·Pablo Mi a3
Recei ed: 15 Ma ch 2024 / Accep ed: 30 Oc obe 2024
© The Au ho (s) 2024
Abs ac
We cons uc , o any H∈R, in ini ely many ee bounda y annuli in geodesic balls o S3
wi h cons an mean cu a u e Hand a disc e e, non- o a ional, symme y g oup. Some o
hese ee bounda y CMC annuli a e ac ually embedded i H≥1/√3. We also cons uc
embedded, non- o a ional, ee bounda y CMC annuli in geodesic balls o H3, o all alues
H>1 o he mean cu a u e H.
Ma hema ics Subjec Classi ica ion 53A10 ·53C42
1 In oduc ion and s a emen o he esul s
1.1 His o ical in oduc ion
Fo a long ime, i was belie ed ha he only closed, i.e., compac wi hou bounda y, su aces
wi h cons an mean cu a u e (CMC) in Euclidean space R3we e he ound, o ally umbilic
sphe es. Hop [16] p o ed in 1951 he alidi y o his s a emen o he pa icula case o
genus ze o. Howe e , he conjec u e was unexpec edly sol ed in he nega i e by Wen e [34]
in 1986, by cons uc ing CMC o i in R3wi h sel -in e sec ions and a disc e e symme y
g oup. Subsequen ly, Ab esch [1]andWal e [30,31] ga e a mo e explici cons uc ion, by
p esc ibing ha he CMC o i we e olia ed by plana cu a u e lines.
A undamen al achie emen o CMC heo y sp inging om hese wo ks was he classi-
ica ion o all CMC o i in he space o ms R3,S3and H3by Pinkall-S e ling [26](inR3),
Hi chin [15] ( o minimal o i in S3) and hen by Bobenko [5] ( o CMC o i in S3and H3),
Communica ed by Manuel del Pino.
BPablo Mi a
[email p o ec ed]
Albe o Ce ezo
ce ezocid@ug .es
Isabel Fe nández
[email p o ec ed]
1Depa amen o de Geome ía y Topología, Uni e sidad de G anada, G anada, Spain
2Depa amen o de Ma emá ica Aplicada I, Ins i u o de Ma emá icas IMUS, Uni e sidad de Se illa,
Se illa, Spain
3Depa amen o de Ma emá ica Aplicada y Es adís ica, Uni e sidad Poli écnica de Ca agena, Ca agena,
Spain
0123456789().: V,- ol 123
37 Page 2 o 44 A. Ce ezo e al.
using algeb o-geome ic me hods om in eg able sys ems. In hese heo ems, CMC o i we e
desc ibed in connec ion wi h ini e ype solu ions o he sinh-Go don equa ion. Roughly, o
any na u al N≥1, he space o ype Nsinh-Go don solu ions is ini e dimensional, all CMC
o i a e o ini e ype, and hey can be de ec ed wi hin hei associa ed ini e dimensional
space by explici closing condi ions.
This classi ica ion did no de ec he possible embeddedness o CMC o i in S3(in R3and
H3 he e a e no closed embedded CMC su aces, by Alexand o ’s heo em). The undamen al
achie emen in his di ec ion was ob ained mo e ecen ly by B endle [6],whop o ed ha he
Cli o d o us is he only embedded minimal o us in S3, he eby sol ing in he a i ma i e a
long-s anding conjec u e by Lawson. The idea in [6] was hen adap ed by And ews and Li
[2] o show ha any embedded CMC o us in S3is o a ional; his sol ed a conjec u e by
Pinkall-S e ling.
The e is a na u al bounda y e sion o he classi ica ion p oblem o CMC o i discussed
abo e: o classi y all ee bounda y CMC annuli in geodesic balls o M3(ε) =R3,S3o H3.
He e, we say ha a compac CMC su ace has ee bounda y in a geodesic ball B⊂M3(ε)
i ⊂Band in e sec s ∂Bo hogonally along ∂. This p oblem was al eady conside ed
by Ni sche [25] in 1985. The solu ions a e c i ical poin s associa ed o a na u al a ia ional
p oblem o he a ea uncional, see [25,28].
Ni sche p o ed in [25] ha any ee bounda y CMC disk in B⊂R3is o ally umbilic. Ros
and Souam [28,29] ex ended his esul o S3and H3.In[25], Ni sche claimed wi hou p oo
ha any ee bounda y CMC annulus in a ball B⊂R3should be o a ional. This claim was
p o ed inco ec by Wen e in 1995 [35], h ough he cons uc ion o imme sed ee bounda y
CMC annuli wi h e y la ge mean cu a u e in he uni ball o R3. Howe e , wo ques ions
emained: he exis ence o non- o a ional ee bounda y minimal annuli and o embedded
ee bounda y CMC annuli in he uni ball.
The i s ques ion was ecen ly answe ed by Fe nández, Hauswi h and Mi a in [11],
whe e hey cons uc ed imme sed ee bounda y minimal annuli in he uni ball. Fo ha ,
hey used he Weie s ass ep esen a ion, and a s udy o minimal su aces in R3wi h sphe ical
cu a u e lines; la e on, Kapouleas-McG a h [17] p esen ed an al e na i e, mo e analy ical
cons uc ion ia doubling. The second ques ion has also been ecen ly answe ed by he
au ho s in [8], by cons uc ing embedded non- o a ional ee bounda y CMC annuli in he
uni ball o R3, h ough a s udy o an o e de e mined sys em o he sinh-Go don equa ion.
This answe ed an open p oblem posed by Wen e in 1995 [35].
These esul s s ill lea e unsol ed he c i ical ca enoid conjec u e, i.e., ha any embedded
ee bounda y minimal annulus in he uni ball o R3should be he c i ical ca enoid. See e.g.
[12,13,20]. This conjec u e is concei ed as he ee bounda y e sion o he a o emen ioned
Lawson conjec u e o minimal o i in S3.
The p oblem o whe he he only embedded ee bounda y minimal annulus in a geodesic
ball o S3is a (sphe ical) c i ical ca enoid has been s udied in wo in e es ing ecen wo ks
by Lima-Menezes [22] and Med ede [23]. In [22], he uniqueness o his c i ical ca enoid
is ob ained among imme sed annuli by imposing ha i s coo dina e unc ions a e i s eigen-
unc ions o a sui able S eklo p oblem. In [23] i is shown ha he Mo se index o he c i ical
ca enoid is 4, ha i s spec al index is 1, and ha any ee bounda y minimal annulus wi h
spec al index 1 is a (sphe ical) c i ical ca enoid. See e.g. [3,7,14,21,24,29,32] o o he
esul s on ee bounda y CMC su aces in sphe ical o hype bolic balls.
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 3 o 44 37
1.2 Main esul s
Ou aim in his pape is o show ha he e exis many non- o a ional ee bounda y CMC
annuli in geodesic balls o S3and H3, which in many cases a e ac ually embedded. In
pa icula , his shows ha he class o minimal annuli in S3conside ed in [22,23] is non-
i ial, and ha he classi ica ion o all ee bounda y CMC annuli in geodesic balls o space
o ms is a e y ich geome ic p oblem, bo h in he imme sed and embedded cases. Ou main
esul s a e desc ibed in Theo ems 1.1 and 1.2 below. They indica e ha :
(1) In S3 he e exis imme sed, non- o a ional ee bounda y H-annuli in geodesic balls
B⊂S3, o anyH∈R. In pa icula , he e exis ee bounda y minimal annuli in S3
wi h a ini e symme y g oup. This answe s a p oblem in [23].
(2) Fo H≥1/√3, some o hese ee bounda y CMC annuli in S3a e embedded.
(3) In H3, he e exis embedded non- o a ional ee bounda y H-annuli in geodesic balls
B⊂H3, o anyH>1.
(4) All hese H-annuli come in 1-pa ame e amilies, and a e olia ed by sphe ical cu a u e
lines. They can be seen as ee bounda y bi u ca ions om ini e co e s o adequa e
o a ional examples (nodoids in H3, ca enoids o nodoids in S3).
We will also show ha he e exis embedded, non- o a ional, capilla y minimal annuli in
geodesic balls o S3.
We should no e ha he analy ic esul s by Kilian and Smi h in [18] p o e ha any ee
bounda y CMC annulus in a geodesic ball o R3,S3o H3is associa ed o a ini e ype solu ion
o he sinh-Go don equa ion. The sphe ical cu a u e lines condi ion o ou examples is e y
na u al in his con ex , since hey co espond o solu ions o ype N=2, see [26,27,33]. The
s udy o CMC su aces in R3wi h sphe ical cu a u e lines da es back o classical wo ks by
di e en ial geome e s o he 19 h cen u y, like Ennepe , Dob ine o Da boux. See [30,31,
33] o mo e mode n app oaches, and [4,9] o s udies on iso he mic su aces wi h sphe ical
cu a u e lines.
In he nex heo ems, we le H≥0andε∈{−1,1}so ha H2+ε>0, and deno e
M3(1)=S3⊂R4and M3(−1)=H3⊂L4.
Theo em 1.1 The e exis s an open in e al J=J(H,ε) con ained in (0,1)such ha ,
o any i educible q =m/n∈J∩Q, he e exis s a eal analy ic 1-pa ame e amily
Fq:= {Aq(η) :η∈[0,
0(q))}o compac annuli in M3(ε) wi h he ollowing p ope ies:
(1) Each annulus Aq(η) has cons an mean cu a u e H, and has ee bounda y in a geodesic
ball B=B(q,η)⊂M3(ε) cen e ed a e4=(0,0,0,1)∈M3(ε).
(2) Aq(η) is symme ic wi h espec o he o ally geodesic su ace S:= M3(ε) ∩{x3=0}.
(3) The closed geodesic S∩Aq(η) o Aq(η) has o a ion index −minS.
(4) Aq(0)is a ini e m-co e o a compac embedded piece o a Delaunay su ace in M3(ε).
(5) I η>0, henAq(η) is no o a ional, and i s symme y g oup is gene a ed by he
symme y wi h espec o S, and by he symme ies wi h espec o n >1equiangula
o ally geodesic su aces o M3(ε) o hogonal o S. Tha is, he symme y g oup o Aq(η)
is p isma ic o o de 4n.
(6) Each annulus Aq(η) is olia ed by sphe ical cu a u e lines, so ha bo h o i s bounda y
componen s a e elemen s o his olia ion.
Theo em 1.2 Assume in Theo em 1.1 ha ε=−1o ha ε=1and H ≥1/√3. Then, he e
exis elemen s in J∩Qo he o m q =1/n, and o any such q, he ee bounda y H-annuli
Aq(η) a e embedded, o ηsu icien ly small. When ε=−1, i ac ually holds 1/n∈J o
any n ≥2.
123
37 Page 4 o 44 A. Ce ezo e al.
1.3 O ganiza ion o he pape and ske ch o he p oo
The s a egy o p o e Theo ems 1.1 and 1.2 is inspi ed by ou p e ious wo ks [8,11]on
ee bounda y CMC annuli in R3. We will s a by cons uc ing a amily o imme sions
ψ(u, ):R2→M3(ε) wi h cons an mean cu a u e H(wi h H>1i ε=−1), depending
on h ee eal pa ame e s (a,b,c), ob ained h ough special solu ions o he sinh-Go don
equa ion. The main p ope y o any such imme sion ψis ha , along each cu e → ψ(u0, ),
i in e sec s wi h a cons an angle θ(u0)some o ally umbilic su ace Q(u0). Mo eo e , i
also sa is ies ψ(−u, ) =(ψ(u, )),whe edeno es he symme y wi h espec o he
o ally geodesic su ace M3(ε) ∩{x3=0}. The p oo o Theo ems 1.1 and 1.2 is hen based
on p o ing ha he e exis pa ame e alues (a,b,c)such ha :
(1) ψis pe iodic in he -di ec ion, and so ψco e s an annulus.
(2) The e exis s u0>0 such ha θ(u0)=π/2andQ(u0)is a sphe e in a ian by .
(3) The annulus ψ([−u0,u0]×S1)⊂M3(ε) lies in he geodesic ball Bo M3(ε) bounded
by Q(u0), and so, because o he p ope ies abo e, i is a ee bounda y annulus in B.
The idea o he p oo will be o bi u ca e om ee bounda y o a ional examples (c i ical
ca enoids o nodoids) wi hin he amily associa ed o he (a,b,c)-pa ame e s, o c ea e some
non- o a ional examples, and o con ol hei embeddedness.
Le us ema k ha he e appea howe e se e al sou ces o complica ion in he p ocess
when we conside S3o H3ins ead o R3as ou ambien space, and hei esolu ion equi es
new ideas. Fo ins ance, we canno use he Weie s ass ep esen a ion o minimal su aces
as in [11]. And, in con as wi h he R3case in [8], in ou sphe ical o hype bolic se ing
we do no ha e an explici exp esion o he pe iod map ha con ols he pe iodici y o
he sphe ical cu a u e lines o ou examples. Mo eo e , we canno use CMC su aces wi h
plana cu a u e lines as a limi in o de o con ol he cen e s o he sphe ical cu a u e
lines, as we did in [8] o he Euclidean case, since such su aces do no exis in S3o H3.
We nex explain he basic s eps o he p oo .
In Sec .2we p esen some p elimina ies on CMC su aces in space o ms olia ed by
sphe ical cu a u e lines, i.e. each cu a u e line o his olia ion lies in some 2-dimensional
o ally umbilic su ace o he ambien space.
In Sec .3we ecall ou cons uc ion in [8] o some special solu ions o an o e de e mined
p oblem o he sinh-Go don equa ion. When his equa ion is iewed as he Gauss equa ion
o a CMC su ace in M3(ε) =S3o H3pa ame ized by cu a u e lines (wi h H>1in
H3), hese solu ions yield comple e CMC su aces in M3(ε) olia ed by sphe ical cu a u e
lines. Along such cu a u e lines, he su ace in e sec s he co esponding o ally umbilic
su ace a a cons an angle, by Joachims al’s heo em. In his way, we end up wi h a amily o
con o mal CMC imme sions ψ(u, ):R2→M3(ε) pa ame ized by cu a u e lines, which
depends on h ee pa ame e s (a,b,c), and so ha he cu es → ψ(u, )a e sphe ical.
In Sec .4we s udy he geome y o he imme sions ψ(u, ). We p o e ha hey a e
symme ic wi h espec o a ho izon al o ally geodesic su ace S⊂M3(ε) and wi h espec
o a numbe o e ical o ally geodesic su aces k⊂M3(ε) ha , in adequa e condi ions,
in e sec along a e ical geodesic o M3(ε) ha con ains all he cen e s o he o ally umbilic
su aces whe e he sphe ical cu a u e lines → ψ(u, )o he imme sion lie.
We also de ine a eal analy ic pe iod map (a,b,c)wi h he p ope y ha , when
(a,b,c)=m/n∈Q, he sphe ical cu a u e lines → ψ(u, ) a e pe iodic cu es
wi h o a ion index m, while ngi es he numbe o e ical symme y su aces k. In pa ic-
ula , when (a,b,c)=m/n, he es ic ion o ψ o [−u0,u0]×Rco e s a CMC annulus
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 5 o 44 37
0=0(a,b,c,u0)in M3(ε) ha in e sec s wi h a cons an angle wo (isome ic) o ally
umbilic su aces o M3(ε).
Thus, ou obje i e will be o show ha along some eal analy ic cu es in he pa ame e
space (a,b,c,u0), he esul ing CMC annuli 0a e ee bounda y in some geodesic ball
B⊂M3(ε). Tha is, we will need o con ol simul aneously ha bo h bounda y cu es o 0
lie in he same o ally umbilic su ace o M3(ε), ha his su ace bounds a compac geodesic
ball B⊂M3(ε), ha 0is o ally con ained in B, and ha he in e sec ion angle along ∂0
wi h ∂Bis π/2. All o his while con olling he possible embeddedness o 0.
To achie e his, he main idea will be o bi u ca e om some o a ional ee bounda y
CMC su ace, so ha he sphe ical cu a u e lines condi ion is p ese ed. Fo his, we will
need a qui e de ailed desc ip ion, ha will be ca ied ou in Sec .5, on he exis ence o c i ical
(sphe ical) ca enoids in S3,andc i ical nodoids in geodesic balls o S3and H3. Sec ion 5is
essen ially independen om he es o he pape , and he p oo s he e a e pos poned o an
appendix.
In Sec .6we will show ha , when a=1, he imme sion ψ(u, )pa ame izes a compac
piece o a o a ional CMC su ace. Mo e speci ically, o ei he a nodoid (in H3), o a nodoid,
a (sphe ical) ca enoid o a la o us in S3. The pa ame e cwill con ol he necksize o his
example. The pa ame e bis needed o accoun o he ac ha , in he o a ional case, each
cu e → ψ(u, )is a ci cle, and so he e is a p io i an in ini e numbe o o ally umbilic
su aces in M3(ε) ha con ain i . The pa ame e bde e mines a choice o such umbilic
su aces.
In Sec .7we will gi e an explici exp ession o he pe iod map (1,b,c)in he case
a=1 ha allows o a good con ol on he pe iodici y o he pa ame ized cu a u e lines
→ ψ(u, ), which in his a=1 case a e me ely ci cles.
In Sec .8we will es ic o a ce ain ee bounda y egion o ou pa ame e space, and
con ol he e, also o he case a=1, he si ua ion in which 0(1,b,c,u0)co e s a c i ical
ca enoid o nodoid in M3(ε). We will show ha hese o a ional ee bounda y su aces mus
appea along ce ain cu es o he pa ame e domain in ha a=1 case.
In Sec .9we p o e Theo ems 1.1 and 1.2. Fo ha we use ou s udy o he o a ional
a=1 case in he p e ious sec ions in wha ega ds he ee bounda y annulus s uc u e o
he examples, and induce i o he non- o a ional a>1 case. The embeddedness is ob ained
o alues o he pe iod (a,b,c)o he o m −1/n, wi h a>1 close o 1.
Finally, in Sec . 10 we use some o he analysis o he p e ious sec ions o cons uc
embedded capilla y CMC su aces in geodesic balls o S3, o all alues o H.
The au ho s a e g a e ul o he e e ee o his pape , o aluable commen s ha helped
imp o ing he exposi ion o he esul s.
1.4 Open p oblems
I is e y na u al o conjec u e ha he (sphe ical) c i ical ca enoids a e he only embedded
ee bounda y minimal annuli in geodesic balls o S3;see[22,23]. Ou esul s imply ha he
embeddedness assump ion canno be emo ed om his conjec u e.
Mo e gene ally, aligned wi h he heo ems in [8,11] o he Euclidean case, one can
conjec u e ha any embedded ee bounda y CMC annulus in a geodesic ball o R3,S3o H3
should be olia ed by sphe ical cu a u e lines. We no e ha he exis ence o an embedded
ee bounda y minimal annulus wi h sphe ical cu a u e lines in a geodesic ball o S3is also
open (in R3, he esul s in [11] show ha hese do no exis ).
123

37 Page 6 o 44 A. Ce ezo e al.
In he imme sed case, i would be in e es ing o cons uc a ee bounda y CMC annulus
in a geodesic ball ha is no olia ed by sphe ical cu a u e lines, o o show ha such an
example canno exis . We also do no know i he e exis con inuous de o ma ions o ee
bounda y CMC annuli in a ixed geodesic ball o R3,S3o H3, wi h a ixed mean cu a u e
H.
2 P elimina ies
Le M3(c0)deno e he space o m o cons an cu a u e c0∈R. In he case c0= 0, we iew
M3(c0)in he usual way as a hype quad ic o R4
ε=(R4,,),
,=dx2
1+dx2
2+dx2
3+εdx2
4,
whe e εis he sign o c0.Tha is,R4
εis ei he he Euclidean 4-space (i ε=1) o he
Lo en z-Minkowski space L4(i ε=−1) and
M3(c0)={x∈R4
ε:x,x=1/c0},
wi h x4>0i ε=−1.
Le deno e an imme sed o ien ed su ace in M3(c0)wi h cons an mean cu a u e
H∈R.Le Ndeno e he uni no mal ec o ield o in M3(c0).Le ζ:= u+i deno e a
con o mal pa ame e o , so ha i s i s undamen al o m is I=e2ω(du2+d 2). Then,
he Codazzi equa ion gi es ha he Hop di e en ial Q:= ψζζ,Nis holomo phic, and he
Gauss equa ion o in he (u, )-pa ame e s is
ω +(H2+c0)e2ω−4|Q|2e−2ω=0.(2.1)
Assume ha is simply connec ed and does no ha e umbilical poin s. Then, a e a
change o con o mal pa ame e , we can assume ha Qis cons an . In ha way, he Gauss
equa ion (2.1)iso he o m
ω +Ae2ω−Be−2ω=0,A,B∈R,B>0.(2.2)
Con e sely, le ω(u, ) :R2→Rsa is y (2.2) wi h espec o cons an s A,B,andle
H,c0,Q∈Rbe cons an s, wi h Q= 0, so ha
A=H2+c0,B=4|Q|2(2.3)
hold. Then, he e exis s an imme sion
ψ(u, ):R2→M3(c0)
wi h cons an mean cu a u e H, Hop di e en ial Q, and whose i s and second undamen al
o ms a e
I=e2ω(du2+d 2), II =(He2ω+2Q)du2+(He2ω−2Q)d 2(2.4)
The p incipal cu a u es κ1>κ
2o ψa e
κ1=H+2|Q|e−2ω,κ
2=H−2|Q|e−2ω.(2.5)
The smalles p incipal cu a u e κ2co esponds o he p incipal u-cu es ( esp. -cu es) i
Q<0 ( esp. Q>0).
This su ace is unique up o o ien a ion p ese ing ambien isome ies, o equi alen ly,
up o p esc ibing he mo ing ame {ψ, ψu,ψ
,N}a some poin (u0,
0). He e, Nis he
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 7 o 44 37
uni no mal o associa ed o he pa ame iza ion ψ. The Gauss-Weinga en o mulas o ψ
a e
ψuu =ωuψu−ω ψ +(He2ω+2Q)N−c0e2ωψ
ψu =ω ψu+ωuψ
ψ =−ωuψu+ω ψ +(He2ω−2Q)N−c0e2ωψ
Nu=−(H+2Qe−2ω)ψu
N =−(H−2Qe−2ω)ψ
(2.6)
We nex analyze he p ope y ha a cu a u e line → ψ(u, ) is sphe ical, i.e. i lies in
a o ally umbilical su ace o M3(c0).
The o ally umbilical su aces o M3(c0)a e gi en by he in e sec ion o hype planes o
R4
εwi h M3(c0). They can be desc ibed by
S[m,d]:={x∈M3(c0):x,m=d},(2.7)
o some m∈R4
ε {0}and d∈R(he e mand da e de e mined up o a common mul iplica i e
ac o ). The condi ion o S[m,d]being a (non-emp y) su ace is ha
m,m−c0d2>0
and ha , i ε=−1andm,m≥0, dand he x4-coo dina e o mha e opposi e signs.
When ε=−1(c0<0),weha e ha S[m,d]is a sphe e ( esp. ho osphe e, pseudosphe e) in
H3(c0)i m,mis nega i e ( esp. ze o, posi i e). I d=0, hen S[m,d]is o ally geodesic
in M3(c0).
Lemma 2.1 Fo each ixed u ∈R, he -cu a u e line ψ(u, )o ψis sphe ical i and only
i he e exis α(u), β(u)∈Rsuch ha
2ωu=α(u)eω+β(u)e−ω.(2.8)
In ha si ua ion, ψin e sec s S[m(u), d(u)]a a cons an angle θ(u)along → ψ(u, )
and we ha e
α=2|
N|Hcos θ−c0d
|
N|sin θ,β=−4Qcos θ
sin θ,(2.9)
whe e |
N|=m,m−c0d2.In pa icula , β=0i and only i cos θ=0, and α=β=0
i and only i cos θ=0and S[m,d]is o ally geodesic in M3(c0).
P oo I o a ixed u=u0 he cu e → ψ(u0, )lies in S[m,d], henm,ψ
=0. One
sees om he e and m,ψ=d ha

N:= m−c0dψ
lies in TM3(c0)and is no mal o S[m,d]along ψ(u0, ),since
N, =0 o e e y ∈R4
ε
o hogonal o bo h ψand m. De ining θby N,
N=|

N|cos θ, one easily ob ains ha ,
changing θby −θi necessa y,
m=e−ω|
N|sin θψ
u+|
N|cos θN+c0dψ. (2.10)
Using his exp ession oge he wi h (2.6)andψ ,m=0, we ob ain (2.8) o α=
α(u0), β =β(u0)gi en by (2.9).
123
37 Page 8 o 44 A. Ce ezo e al.
Con e sely, assume (2.8) holds along he line → (u0, ). Then, (2.8) oge he wi h
(2.6), imply ha he exp ession
m=4Qe−ωψu−βN−(2Qα+Hβ)ψ (2.11)
sa is ies m (u0, ) =0, i.e., mis cons an along → (u0, ). The e o e, m,ψis also
cons an along → (u0, ), i.e., he cu a u e line ψ(u0, )is sphe ical. 
3 Special solu ions o he sinh-Go don equa ion
In his sec ion we ecall he cons uc ion in ou p e ious pape [8] o some special solu ions
o he ellip ic sinh-Go don equa ion, ha will be used la e on. We will make a mo e de ailed
discussion han in [8], in o de o mo i a e hei o igin. We will also indica e some new
addi ional p ope ies o hese solu ions ha will be impo an o ou pu poses he e.
3.1 Wen e’s o e de e mined sys em
We will seek solu ions ρ(u, ):R2→R o he o e de e mined sys em
ρ +coshρsinh ρ=0,(3.1)
2ρu=α(u)eρ+
β(u)e−ρ.(3.2)
o unc ions α, 
β:R→R. No e ha he sys em (3.1)–(3.2) is p ecisely he sys em (2.2)–
(2.8) o he choices A=B=1/4. We will w i e (3.2) in he al e na i e o m
ρu=y(u)coshρ+z(u)sinh ρ, (3.3)
whe e y(u), z(u)a e eal unc ions, so ha α=y+zand 
β=y−z. The nex discussion
is aken om Wen e [33], pages 9-11 and 16-18.
To s a , assume ha ρ(u, )sa is ies (3.1)and(3.3). Then, y(u), z(u)should be a solu ion
o he di e en ial sys em
y =(a−1)y−2y(y2−z2),
z =az −2z(y2−z2), (3.4)
wi h espec o some cons an a, see equa ion (3.6)in[33].
Mo eo e , i we deno e Z(u, ):= eρ(u, ), hen i ollows by a compu a ion om equa ion
(2.20) in [33] (choosing A=B=1/4 and making he change α=y+z,β=y−zas
explained abo e) ha
4Z2
=p(u,Z), (3.5)
whe e
p(u,x):= −(1+(y+z)2)x4−4(y+z)x3+6γx2+4(y−z)x−(1+(y−z)2).
(3.6)
He e, we a e deno ing y=y(u),z=z(u),and6γ=6(y2−z2)−4(a−1/2). Thus, o
each ixed u alue, p(u,x)is a polynomial o deg ee ou .
This p ocess can be e e sed unde some addi ional condi ions, as explained in [33,Thm.
2.3]. Speci ically, assume ha we p esc ibe he ini ial alues
(y(0), z(0), y(0), z(0),a,ρ(0,0)). (3.7)
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 9 o 44 37
so ha p(0,eρ(0,0))>0, whe e p(u,x)is as in (3.6). Then, by [33, Thm. 2.3], he e exis s
a solu ion ρ(u, ) o (3.1)–(3.3) whose associa ed unc ions y(u), z(u)sol e (3.4) o he
gi en cons an a, wi h he gi en ini ial condi ions y(0), z(0), y(0), z(0).
The sys em (3.4) has a Hamil onian s uc u e. The basic Hamil onian cons an s o (3.4)
a e
y2−z2−(a−1)y2+az2+(y2−z2)2=h∈R(3.8)
and
(zy−yz)2+z2+z2(y2−z2−a)=k∈R,(3.9)
see (3.7)in[33].
One can hen ollow he classical p ocedu e o sol e he Hamil on-Jacobi equa ions by
sepa a ion o a iables in o de o w i e sys em (3.4) in o a mo e adequa e o m; see [33, pg.
17]. Speci ically, i s we apply he change o a iables
y2=−(1−s)(1− ), z2=−s .(3.10)
Using (3.10), he sys em (3.4) ans o ms in o he au onomous i s o de sys em
s(λ)2=s(s−1)q(s), (s≥1),
(λ)2= ( −1)q( ), ( ≤0), (3.11)
whe e q(x)is he hi d-deg ee polynomial
q(x)=−x3+(a+1)x2+(h−a)x+k,(3.12)
and he pa ame e λo (3.11) is linked o uby (see [33, pg. 18])
2u(λ) =s(λ) − (λ) > 1.(3.13)
3.2 Cons uc ions o special solu ions o he sys em
We now explain ou cons uc ion in [8]. To s a , we will ix
y(0)=z(0)=0 (3.14)
by geome ical easons. Mo e speci ically, we in end o use (3.1) as he Gauss equa ion
o a CMC su ace in some space M3(c0), see Sec .4below. These ini ial condi ions
will de e mine ha in e sec s o hogonally a o ally geodesic su ace o M3(c0). See also
Lemma 2.1. In pa icula will ha e a use ul symme y plane.
We a e in e es ed in he possibili y o ob aining embedded examples o ee bounda y
CMC annuli in geodesic balls o M3(c0). A e ixing he ini ial condi ions (3.14), a leng hy,
de ailed inspec ion o all he cases ha we will no ep oduce he e seems o indica e ha his
embeddedness is only possible when p(0,x)in (3.6) has wo posi i e oo s and wo nega i e
oo s. So, we should p esc ibe he alues o y(0), z(0)andain a way ha his p ope y holds
o p(0,x)in (3.6). We will also be in e es ed in he si ua ion whe e he wo posi i e oo s
o p(0,x)collapse in o one, since his si ua ion will de ec Delaunay examples in M3(c0).
Because o his, i seems con enien o seek ini ial condi ions y(0), z(0),aso ha p(0,x)
in (3.6) can be w i en in he ac o o m below, which will o ce i o ha e he desi ed oo
s uc u e:
p(0,x)=−x−a
cx−1
ac(x+bc)x+c
b,(3.15)
123
37 Page 16 o 44 A. Ce ezo e al.
Le ϕbe he s e eog aphic p ojec ion o M3(ε) om −e4.So,i ε=1, ϕmaps S3 {−e4}
in o R3, while i ε=−1, ϕmaps H3in o he uni ball B3⊂R3.Sincelies in {x3=0},
he cu e γ:= ϕ◦is hen a plana cu e in {z=0}⊂R3,whe e(x,y,z)deno e he
Euclidean coo dina es o R3.
De ini ion 4.8 Using he abo e no a ion, we de ine he pe iod map as
:O−→R,
(a,b,c):= 1
πσ
0
κγ||γ||d , (4.16)
whe e κγand ||γ|| deno e he Euclidean cu a u e and he leng h o γ, espec i ely. No e
ha π ep esen s he a ia ion o he (Euclidean) uni no mal o γ( ) along he in e al
[0,σ].
P oposi ion 4.9 The map =(a,b,c)in (4.16) is eal analy ic in O−.
P oo I is an immedia e consequence o he eal analy ici y o σ=σ(a,b,c)(P opo-
si ion 3.2)andρ=ρ(u, ;a,b,c), oge he wi h he analy ic dependence o he
Gauss–Weinga en sys em (4.3) wi h espec o ini ial condi ions. 
We explain nex he geome y o when i s associa ed pe iod is a a ional numbe . We
s a by desc ibing he geome y o he plana geodesic ( ) =ψ(0, ).
P oposi ion 4.10 Assume ha (a,b,c)=m/n∈Q, wi h n ∈N {0}and m/n i educible.
Then ( +2nσ) =( ). In pa icula ( ) is a closed cu e.
Mo eo e , ( ) has o a ion index m ∈Zand, i a >1, a dihed al symme y g oup Dn
wi h n ≥2.
P oo Le us see i s ha γ=ϕ◦sa is ies γ( +2nσ) =γ( ), whe e as be o e ϕdeno es
he s e eog aphic p ojec ion o M3(ε) om −e4. Conside he o ally geodesic su aces
k=Pk∩M3(ε) (see P oposi ion 4.5). Then, ϕmaps kin o e ical planes kcon aining
he z-axis. Le Lk:= k∩{z=0}.Then{Lk:k∈Z}is a amily o equiangula lines in
{z=0}≡R2passing h ough he o igin. I ollows om P oposi ion 4.5 and he ac ha
ϕis con o mal ha he angle be ween Lkand Lk+1is he angle be weeen γ(0)and γ(σ ).
We deno e his angle by ϑ.No e ha γ(kσ)is o hogonal o Lk. This implies ha
π =2πl+ϑ(4.17)
o some l∈Z. In pa icula , nϑ∈πZ.
By (4.13), we ha e
γ(kσ− ) =Tk(γ (kσ+ )), (4.18)
whe e Tkdeno es he symme y o R2 ha ixes Lk.I Rdeno es he o a ion a ound he
o igin o angle 2ϑ, henweha eby(4.18) ha γ( +2σ) =R(γ ( )).F omhe eand
nϑ∈πZwe ob ain γ( +2nσ) =γ( ).
Also om (4.18) we ob ain ha := ||γ||κγsa is ies (kσ− ) = (kσ+ ) o all
k∈Z. Obse e ha his implies ha , o all k∈Z,
=1
π(k+1)σ
kσ
κγ||γ||d .
123

F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 17 o 44 37
F om he e, =m/nand he 2σn-pe iodici y o γ( ), i ollows ha he o a ion index o
γ( )is equal o m.
Finally, we de e mine he symme y g oup o ( ) when a>1. In ha case we know ha
X( ) in (3.23) only has c i ical alues a he poin s o he o m kσ,k∈Z. Also, by (4.4)and
since ψ(u, )in e sec s S=M3(ε) ∩{x3=0}o hogonally along ( ),weha e
κ( ) =κ2(0, )=H−μ
X( )2,
whe e κis he geodesic cu a u e o in S. Since he s e eog aphic p ojec ion ϕp ese es
he c i ical poin s o he geodesic cu a u e o egula cu es (because i p ese es cu es
o cons an cu a u e, and hence he con ac o de wi h hese cu es), we deduce ha κγ( )
only has c i ical poin s a he alues =kσ,k∈Z.
In pa icula , γ( )has a ( ini e) dihed al symme y g oup, as i is symme ic wi h espec
o he e lec ions T1,...,Tn. So, i s isome y g oup is Dn o some n≥n,sincem/nis
i educible. I n>n, he e would exis some addi ional symme y line L o γ( )di e en
om all Lk.So,( 0− ) =(( 0+ )) o some 0/∈{kσ:k∈Z},whe e
is he symme y wi h espec o L. Thus, κγwould ha e a c i ical poin a 0,wha isa
con adic ion. Hence, he symme y g oup o γ( ) is Dn, and gene a ed by he e lec ions
T1,...,Tn. The e o e, he symme y g oup o ( ) is isomop hic o Dn, as claimed.
Finally, we show ha n≥2. Indeed, i n=1, hen all he Lk’s ag ee, and his con adic s
ha (a,b,c)∈O−,since{ν0,ν
1}would be collinea . 
As an immedia e consequence o P oposi ion 4.10,weha e:
Co olla y 4.11 Le (a,b,c)∈O−so ha (a,b,c)=m/n∈Q,whe en∈N {0}, wi h
m/n i educible. Then, ψ(u, +2nσ) =ψ(u, ).
4.5 Cons uc ion o CMC annuli
Following he esul s in Sec . 4.4,gi enu0>0, we can de ine 0=0(a,b,c,u0)as he
es ic ion o ψ(u, ) o [−u0,u0]×R. By Co olla y 4.11,i (a,b,c)=m/n∈Q, we can
iew 0as a compac H-annulus in M3(ε) unde he iden i ica ion (u, +2nσ) ∼(u, ).
Wi h his, we ha e ou main conclusion o his sec ion:
Theo em 4.12 Le (a,b,c)∈O−so ha (a,b,c)=m/n∈Q,whe en ∈N {0},
wi h m/n i educible. Then, o any u0>0, he ollowing p ope ies hold o he annulus
0=0(a,b,c,u0):
(1) 0is symme ic wi h espec o S=M3(ε) ∩{x3=0}, and wi h espec o n ≥2 o ally
geodesic su aces 1,...,
no M3(ε) ha in e sec equiangula ly along a geodesic
Lo M3(ε) o hogonal o S.
(2) Along each bounda y componen ∂i
0,i =1,2,0in e sec s a a cons an angle θa
o ally umbilic su ace Qio M3(ε). Speci ically, he in e sec ion angle θis he same a
bo h componen s, and Q1=(Q2),whe eis he symme y o M3(ε) wi h espec o
S.
(3) Assume u0=τ,whe eτ=τ(a,b,c)is de ined in P oposi ion 3.4.Thenθ=π/2, i.e.,
∂i
0in e sec s Qio hogonally.
(4) Assume ha m3(u0)=0,whe em
3deno es he x3-coo dina e o he cen e map m in
(2.10). Then bo h bounda y cu es ∂1
0,∂2
0lie in he same o ally umbilic 2-sphe e
Q1=Q2o M3(ε).
123
37 Page 18 o 44 A. Ce ezo e al.
(5) I a >1, he symme y g oup o 0is isomo phic o Dn×Z2, and gene a ed by he
symme ies in i em (1). In pa icula , 0is no o a ional.
P oo I em (1) is a di ec consequence o P oposi ion 4.10. I em (2) ollows om he sym-
me y o 0wi h espec o Sand he ac ha he bounda y cu es o 0co espond o he
sphe ical cu a u e lines ψ(±u0, ).
I em (3) ollows om he de ini ion o τin P oposi ion 3.4, oge he wi h (4.5)and(2.9).
Rega ding i em (4), we i s no e ha S[m(u0), d(u0)]is one o Q1,Q2; he e, we ollow
he no a ion o Lemma 2.1.I m3(u0)=0, he cen e o his Qilies in {x3=0}.Since
Q1=(Q2)by i em (2), we ha e Q1=Q2=S[m(u0), d(u0)]. We show nex ha
S[m(u0), d(u0)]is an umbilic 2-sphe e i ε=−1, a p ope y equi alen o m(u0)being
imelike by he discussion be o e Lemma 2.1.Le Pdeno e he plane whe e m(u)lies, as
speci ied in P oposi ion 4.3.Since(a,b,c)∈O−,Pis imelike, and by P oposi ion 4.5 i
con ains e3.Sincem(u0), e3=0, hen m(u0)is imelike, as desi ed.
To p o e i em (5), le deno e an isome y o M3(ε) ha lea es 0in a ian . Unde his
isome y, we mus ha e () =, since he poin s o  ep esen he middle poin s o he
(in insic) geodesics j∩0o 0. The e o e, he es ic ion o  o Sis a symme y o ,
and hence a composi ion To he symme ies Tjwi h espec o some j, by P oposi ion 4.10.
I  akes each bounda y componen o 0 o i sel , we deduce hen ha =T. O he wise
=◦T,whe eis he symme y wi h espec o S(no e ha in e changes he bounda y
componen s o 0). This p o es i em (5). 
Theo em 4.12 mo i a es he nex de ini ion and consequence.
De ini ion 4.13 Fo any (a,b,c)∈W,wede ineh:W→Ras he map h(a,b,c):=
m3(τ(a,b,c)),whe em3deno es he x3-coo dina e unc ion o he cen e m(u).
Co olla y 4.14 Le (a,b,c)∈O−∩Wso ha (a,b,c)=m/n∈Q,whe en∈N {0}.
Assume ha h(a,b,c)=0. Then, choosing u0=τ, bo h bounda y componen s o he
annulus 0in Theo em 4.12 in e sec o hogonally he same umbilic 2-sphe e Qo M3(ε).
5 C i ical ca enoids and nodoids in space o ms
In his sec ion we in oduce he amily o o a ional CMC su aces in M3(ε) =S3o H3
ollowing do Ca mo-Dajcze [10], and p o e ha each elemen wi hin a sub amily o hem
has a compac piece ha is an embedded ee bounda y o a ional annulus in an adequa e
geodesic ball o M3(ε). This sec ion can be ea ed independen ly o he es o he pape ;
he p oo s a e pos poned o an appendix.
Up o an isome y, any o a ional imme sion in M3(1)=S3can be exp essed as
ψ(s,θ)=(x(s)cos θ,−x(s)sin θ,1−x(s)2sin(φ(s)), 1−x(s)2cos(φ(s)))).(5.1)
o some unc ions x=x(s),φ=φ(s).
In he case o M3(−1)=H3, we will ocus on o a ional su aces o ellip ic ype, ha is,
su aces ha a e in a ian by a compac , con inuous 1-pa ame e subg oup o isome ies o
H3. Up o an isome y, any o a ional su ace o his ype can be exp essed as
ψ(s,θ)=(x(s)cos θ,−x(s)sin θ,x(s)2+1sinh(φ(s)), x(s)2+1cosh(φ(s)). (5.2)
I he imme sion has CMC H≥0 and is no o ally umbilic, and we choose sas he a cleng h
pa ame e o i s p o ile cu e, i can be shown ha x(s)is an analy ic unc ion sa is ying he
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 19 o 44 37
ollowing di e en ial equa ion:
x2=h(x)
x2:= x2−εx4−(Hx2−δ)2
x2,(5.3)
o some cons an δ= 0. In o de o (5.3) o ha e solu ions, i is necessa y ha he biquad a ic
polynomial h(x)be non-nega i e o some x∈R. We will ea he cases ε=1andε=−1
sepa a ely.
5.1 Ro a ional CMC su aces in S3
In his case, he polynomial h(x)in (5.3) is non-nega i e o some x∈Ri and only i
δ∈H−μ
2,H+μ
2(5.4)
whe e, as in he p e ious sec ion, μ:= √H2+1.
I δ=H±μ
2 hen h(x)is non-posi i e, wi h double oo s a he alues x=±
√|δ|/μ.
In his case, he only solu ions x(s)o (5.3) a e he cons an ones, x(s)≡±
√|δ|/μ (up
o an isome y, we can assume ha x(s)≡√|δ|/μ). Fo δ∈(H−μ
2,H+μ
2),δ= 0, h(x)
has ou simple oo s {−xM,−xm,xm,xM}, wi h 0 <xm<xM≤1, and h(x)≥0 o
all x∈[−xM,−xm]∪[xm,xM]. In his case, (5.3) has wo ypes o analy ic solu ions: he
cons an ones, and he non-cons an ones, which oscilla e ei he on he in e al [−xM,−xm]
o on [xm,xM]. The only solu ions ha gi e ise o CMC imme sions in S3a e he non-
cons an ones. Up o isome ies, we can always assume ha x(0)=xm.
P oposi ion 5.1 ([10]) Le ε=1,H≥0and δ= 0such ha (5.4) holds. I δ=H±μ
2,le
x(s):= √|δ|/μ, cons an . O he wise, le x(s)be he unique non-cons an solu ion o (5.3)
wi h ini ial condi ion x(0)=xm.Wede ine
φ(s):= s
0
δ−Hx2
x(1−x2)ds.(5.5)
Then, deno ing S1≡R/(2πZ), he imme sion ψ:R×S1→S3gi en by (5.1) de ines a
o a ional su ace in S3wi h cons an mean cu a u e H ≥0such ha ψs,ψ
s≡1.The
s-cu es and θ-cu es a e cu a u e lines, wi h espec i e associa ed p incipal cu a u es
κs=H+δ/x2,κ
θ=H−δ/x2.(5.6)
Con e sely, any o a ional CMC su ace in S3mus be an open piece o ei he one o hese
examples, o o a o ally umbilical ound sphe e.
De ini ion 5.2 (Sphe ical nodoids, unduloids and ca enoids) Le ε=1. Fo any H≥0and
δ= 0 such ha (5.4) holds, le S=S(ε, H,δ) be he o a ional CMC su ace in S3o
P oposi ion 5.1.
•I H>0and0<δ<(H+μ)/2 we will say ha Sis a sphe ical nodoid.
•I H>0and(H−μ)/2<δ<0 we will say ha Sis a sphe ical unduloid.
•I H=0 we will say ha Sis a sphe ical ca enoid. Since he change δ→−δjus gi es
a epa ame e iza ion o S, we can assume in his case ha δ>0.
•I δ=(H±μ)/2, he su ace co e s a la o us. Fo H=0, i is a Cli o d o us.
123
37 Page 20 o 44 A. Ce ezo e al.
5.2 Ro a ional CMC su aces in H3o ellip ic ype
We ecall ha , in o de o (5.3) o ha e solu ions, i is necessa y ha h(x)be non-nega i e
o some x∈R. This will happen in any o he ollowing si ua ions:
(1) H<1,
(2) H=1andδ>−1
2,
(3) H>1andδ≥μ−H
2,whe eμ:= √H2−1.
In he i s wo cases, h(x)only has wo oo s −xm<0<xm,andh(x)≥0 o allx∈
(−∞,−xm]∪[xm,∞). In he hi d case, h(x)has ou oo s −xM≤−xm<0<xm≤xM,
and xm=xMi and only i δ=μ−H
2. In his si ua ion, h(x)≥0on[−xM,−xm]∪[xm,xM].
I H>1andδ=μ−H
2, henxm=xM=H−μ
2and he only analy ic solu ions o (5.3)
a e he cons an s x=±xm. The esul ing su aces a e la hype bolic cylinde s in H3.
In he es o cases, (5.3) has wo ypes o analy ic solu ions: he cons an s gi en by he oo s
o h(x), and non-cons an solu ions. The only ones ha gi e ise o ac ual CMC imme sions
a e hose which a e no cons an . Mo e speci ically, i H>1, hen x(s):R→Roscilla es
on ei he [−xM,−xm]o [xm,xM].I H≤1, howe e , x(s):R→Ris unbounded, aking
alues on ei he (−∞,−xm]o [xm,∞).
In all h ee cases, up o isome ies in H3, we can always assume ha x(0)=xm,whe e
xmis he smalles posi i e oo o h(x).Weha e hen:
P oposi ion 5.3 ([10]) Le ε=−1,H≥0,δ= 0.I H >1and δ=μ−H
2,le x(s)≡xm.
O he wise, le x(s)be he unique noncons an solu ion o (5.3) wi h ini ial condi ion x(0)=
xm.Wede ine
φ(s):= s
0
δ−Hx2
x(x2+1)ds.(5.7)
Unde hese condi ions, he imme sion ψ:R×S1→H3in (5.2) is a o a ional su ace in
H3wi h cons an mean cu a u e H ≥0and ψs,ψ
s≡1, wi h p incipal cu a u es gi en
by (5.6).
Con e sely, any o a ional CMC su ace o ellip ic ype in H3mus be an open piece o
ei he one o hese examples, o o a o ally umbilical su ace o H3.
De ini ion 5.4 We will say ha he imme sion ψin P oposi ion 5.3 is a hype bolic nodoid
( esp. unduloid)i H>1andδ>0 ( esp. μ−H
2<δ<0), and deno e i by S=S(ε, H,δ).
5.3 Exis ence o ee bounda y nodoids and ca enoids
Ou goal in his sec ion is o show ha he nodoids and ca enoids gi en in P oposi ions 5.1,5.3
o δ>0 (see also De ini ions 5.2 and 5.4) a e ee bounda y in a ce ain ball o M3(ε).The
geodesic balls o M3(ε) will be desc ibed as
B[m,d]:={x∈M3(ε) :x,m≥d}.(5.8)
He e m∈M3(ε) is he cen e o he ball, while d∈Rsa is ies |d|<1whenε=1and
d<−1whenε=−1.
The ac ha x(0)=xmand φ(0)=0 along wi h (5.3), (5.5), (5.7) imply ha he unc ion
x(s)is symme ic, while φ(s)is an isymme ic. A geome ic consequence o his is ha he
imme sions ψ(s,θ)gi en in (5.1), (5.2) a e symme ic wi h espec o he o ally geodesic
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 21 o 44 37
su ace S={x3=0}∩M3(ε). Mo eo e , he o a ion axis o hese examples is gi en by he
geodesic L:= {x1=x2=0}∩M3(ε). We no e ha he balls B[e4,d]cen e ed a he poin
e4∈M3(ε) a e also symme ic wi h espec o Sand in a ian unde o a ions wi h axis L.
Gi en s0>0, we de ine S0as he compac annulus ψ([−s0,s0]×S1)⊂M3(ε),whe e
we ha e iden i ied he poin s (s,θ +2π) ∼(s,θ)in he ob ious way. Conside he p o ile
cu e o S0,
s→ ψ(s,0)=(x(s), 0,x3(s), x4(s)). (5.9)
We a e in e es ed in s udying whe he S0is ee bounda y in a ce ain ball. By he symme ies
o S0, i can be shown ha i s wo bounda y componen s a e con ained in he o ally umbilical
sphe e
S[e4,εx4(s0)]=∂B[e4,εx4(s0)]⊂M3(ε).
So, i S0happened o be ee bounda y in some geodesic ball o M3(ε),i wouldnecessa ily
be in B[e4,εx4(s0)].
P oposi ion 5.5 Le ε∈{−1,1},H≥0,δ>0.I ε=1, assume u he ha δ< H+μ
2.Fo
any s, deno e by s hegeodesicinM3(ε) wi h ini ial condi ions ψ(s,0)and ψs(s,0).
(1) The e exis s
s=
s(H,δ) > 0such ha he compac annulus S0:= ψ([−
s,
s]×S1)is
embedded and ee bounda y in he ball B := B[e4,εx4(

s)]⊂M3(ε), wi h x4(

s)>0.
Mo eo e , he p incipal cu a u e associa ed o he p o ile cu e is s ic ly dec easing
on a ce ain in e al [0,
s+),>0.
(2) The map (H,δ) →
s(H,δ) is analy ic. Mo eo e , i ε=1,
sex ends con inuously o
he bounda y cu e δ=H+μ
2by

sH,H+μ
2=π
2√2μ(H+μ) .(5.10)
(3) On a neighbou hood I⊂Ro 
s=
s(H,δ), he o a ional axis Lo S0and he geodesic
smee a a unique poin p(s)wi h p4(s)>0. Mo eo e , he unc ion p:I→Lis
analy ic wi h p3(

s)=0and p3(

s)>0(he e p3(s), p4(s)deno e he hi d and ou h
coo dina es o p(s) espec i ely).
P oo See Appendix A.
Rema k 5.6 The i s i em o he p e ious P oposi ion omi s he limi case δ=H+μ
2,which
co esponds o o a ional la o i. Howe e , in his case he unc ions x(s),x3(s)can be
compu ed explici ly:
x(s)≡μ+H
2μ,x3(s)=μ−H
2μsin 2μ(μ +H)s.(5.11)
Wi h his, one can check ha he embedded annulus ψ([−
s,
s]×S1),whe e

s=π
2√2μ(μ+H),
is ee bounda y in B[e4,0].
6 Delaunay su aces ia double oo s
We now conside he case a=1 in ou discussion o Sec .4, i.e., we choose (1,b,c)∈O
and le =(1,b,c)be he CMC su ace in M3(ε) o De ini ion 4.1. This means ha
123

37 Page 22 o 44 A. Ce ezo e al.
p(0,x)in (3.15) has a double oo a x=1/c. As explained a e (3.24), in his case we
ha e eρ(0, ) =1/c. The e o e, ρ (0, )≡0 and by uniqueness o he solu ion o he Cauchy
p oblem o (3.1)weha e ha ρ=ρ(u), i.e., ρ ≡0. I ollows hen by (4.2) ha he
coe icien s o I,II o only depend on u,andsois in a ian unde a 1-pa ame e
g oup o ambien isome ies o M3(ε). Mo eo e , each cu e → ψ(u, ) is an o bi o
such 1-pa ame e g oup.
In ou si ua ion, ( ) := ψ(0, )is such an o bi , which ac ually lies in he o ally geodesic
su ace S:= M3(ε) ∩{x3=0}o M3(ε), see Lemma 4.4.Sincein e sec s So hogonally
along ( ), he geodesic cu a u e o ( ) as a cu e in Sis gi en by he p incipal cu a u e
κ2(0, ), which by (4.4) is cons an , and equal o H−μc2.
The e o e, in he case ε=1, is a o a ional CMC su ace in S3.Theu-cu es ( esp.
-cu es) o co espond o he s-cu es ( esp θ-cu es) in he pa ame iza ion ψ(s,θ)o
o a ional su aces o Sec .5.In hisway,ψ(u,0)isap o ilecu eo .
The p incipal cu a u e κ1(u)associa ed o his p o ile cu e ψ(u,0)is always posi i e,
by (4.4). Thus, i H= 0, is he uni e sal co e o ei he a la o us o a sphe ical nodoid
o S3.I H=0, co e s a sphe ical ca enoid o a Cli o d o us. See Sec .5.
Rema k 6.1 I a=c=1, hen y(0)=0by(3.20), and so y(u)≡0. This implies by (3.21)
and (3.3) ha ρ(u)≡0. Thus, a=c=1 co esponds o he case whe e co e s a la
CMC o us in S3.
In he case ε=−1, we ha e ha is a gene alized o a ional su ace in H3, i.e., i is in a ian
by ei he hype bolic, ellip ic o pa abolic o a ions in H3. We will be in e es ed in he ellip ic
case, i.e. he case whe e he o bi s o hese o a ions a e (compac ) ci cles. This happens i
and only i he geodesic cu a u e o he o bi s is g ea e han 1 in absolu e alue, ha is, i
and only i
(H−μc2)2>1.(6.1)
Thus, i (6.1) holds o ou choice o (H,c), henwill be a Delaunay su ace in H3wi h
cons an mean cu a u e H>1. Again, since κ1is posi i e by (4.4) and desc ibes he
geodesic cu a u e o he p o ile cu e o , we deduce ha is a hype bolic nodoid.See
Sec .5. An equi alen o m o (6.1)is
H<μ
2c2+1
c2,(6.2)
whe eweha eused ha μ2=H2−1i ε=−1. This mo i a es he ollowing de ini ion.
De ini ion 6.2 We le R⊂R3be he open subse o O∩{a=1}gi en by
R:= {(1,b,c)∈O:(H−μc2)2+ε>0}.
No e ha Ris jus O∩{a=1}i ε=1, and ha he inequali y de ining Ris (6.1)i
ε=−1. We hus ha e om he discussion abo e:
P oposi ion 6.3 Assume ha (1,b,c)∈R. Then, =(1,b,c)is he uni e sal co e o
he (sphe ical o hype bolic) nodoid (H = 0) o ca enoid (H =0)inM3(ε) wi h neck
cu a u e gi en by κ=H−μc2.
Rema k 6.4 I (1,b,c)∈R∩O−, hen he o a ion axis o (1,b,c)is he geodesic L
desc ibed in i em (1) o Theo em 4.12. This ollows om he ac ha , in his case, ( ) =
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 23 o 44 37
ψ(0, )is a (compac ) ci cle, and any o he symme ies in ha i em (1) mus lea e he cen e
o ( ) ixed, i.e. he cen e lies in he in e sec ion o he symme y planes k. Since bo h
he o a ion axis and ka e o hogonal o S, we conclude om he e ha he o a ion axis
ag ees wi h L=∩
k∈Zk.
I ollows om he abo e cons uc ion ha he pa ame e cde e mines uniquely he imme -
sion ψ(u, ) ha de ines (1,b,c). On he o he hand, he ole o he pa ame e bis o
de e mine he ini ial alues (3.7)o sys em(3.4) ia(3.19), (3.20). Mo e concep ually, since
ψ(u, ) is o a ional, each cu e → ψ(u, ) is a ci cle ha can be seen as con ained in
in ini ely many 2-dimensional o ally umbilic su aces S[m(u), d(u)]o M3(ε). The choice
o bin (1,b,c)de e mines he alues o m(u), d(u)in his desc ip ion.
We will nex desc ibe he beha io o he solu ions o sys em (3.4) associa ed o ou
solu ion ρ(u, ) in ou cu en case a=1. So, we conside he ini ial condi ions (3.14),
(3.19), (3.20) o sys em (3.4), ha gi e oge he wi h (3.21) he solu ion ρ(u, ) o (3.1)
s a ing om (1,b,c)∈O−cons uc ed in Theo em 3.1.
The Hamil onian cons an s h,kin (3.8)and(3.9) can be exp essed in e ms o (b,c)using
a=1, (3.19)and(3.20). This le s us w i e q(x)in (3.12) in e ms o (b,c)as
q(x)=−(x− 1)2(x− 3), (6.3)
whe e
1=−(b−1)2
4b≤0(6.4)
and
3=1
4c+1
c2
≥1.(6.5)
Le (s(λ), (λ)) be he solu ion o (3.11) ob ained a e he change o coo dina es (3.10)
om ou ini ial solu ion (y(u), z(u)) o (3.4). We lis he ollowing p ope ies, ha we e
ob ained in [8], and ha will be used la e on:
i) (s(λ), (λ)) is de ined o all λ∈R, and up o a ansla ion in he λpa ame e i sa is ies
he ini ial condi ions s(0)=1, (0)=0.
ii) s(λ) akes aluesin[1, 3], while (λ) akes alues in [ 1,0]. Mo eo e , s(λ) ≡1i and
only i 3=1and (λ) ≡0 i and only i 1=0.
iii) s(λ) is 2l-pe iodic, whe e
l= 3
1
dx
√x(x−1)q(x)<∞.
In his way, s(2l)=s(0)=1ands(l)= 3.
i ) I 1<0, hen (λ) is s ic ly dec easing, wi h (λ) → 1as λ→∞.
7 The pe iod map o nodoids and ca enoids
In his sec ion we will assume, as in Sec .6, ha a=1, and keep he same no a ions.
The e o e, ρ=ρ(u),andis he o a ional example o P oposi ion 6.3. In pa icula
κ2(0, )=H−μc2.
As explained in Sec .6,i ε=1, ( ) is a ci cle in S3∩{x3=0}. Also, i ε=−1, ( ) is
acu einH3∩{x3=0}≡H2o cons an cu a u e H−μc2. This cu e will be a (compac )
123
37 Page 24 o 44 A. Ce ezo e al.
ci cle i and only i (6.1) holds, i.e., i and only i (1,b,c)∈R(see De ini ion 6.2). We
also ecall ha he se O−and he pe iod map we e in oduced in De ini ions 4.6 and 4.8
espec i ely.
We p o e nex :
Theo em 7.1 Le (1,b,c)∈R∩O−. Then,
(1,b,c)=−(H−c2μ)2+ε
cμ1
c+bc1
c+c
b.(7.1)
P oo Conside he plana cu e γ( ) =ϕ(( )) de ined abo e (4.16). The me ic on
S {−e4}can be w i en ia he in e se s e eog aphic map ϕ−1as
!(dx2+dy2), ! := 4
(1+ε(x2+y2))2,(7.2)
whe e (x,y)a e Euclidean coo dina es in R2.F omhe eand(4.2), we ha e
||γ|| = X
2μ√!,(7.3)
whe e we ha e used ou usual no a ion X( ) =eρ(0, ). On he o he hand, by s anda d
o mulas o con o mally ela ed Riemannian me ics, he geodesic cu a u e κo ( ) in S,
i.e., he geodesic cu a u e o γ( )wi h espec o he me ic (7.2), is ela ed o he Euclidean
geodesic cu a u e κγo γby
κγ=∇√!, n
√!+√!κ,(7.4)
whe e nis he uni no mal o γin R2, and bo h ∇,,a e Euclidean. Recall ha κ=
κ2(0, )=H−μc2<0. Since ( ) is a ho izon al ci cle, i s s e eog aphic p ojec ion γ( )
pa ame izes a ci cle o a ce ain adius >0 in he plane (o in he uni disk o R2,i
ε=−1), which is nega i ely o ien ed since κ<0. Along γ( )we ha e
√!=2
1+ε 2,∇√!=−4ε
(1+ε 2)2γ( ), (7.5)
and n=1
γ( ). Then, i ollows om (7.4), (7.5)andκγ=−1/ ha
=εH−c2μ±(H−c2μ)2+ε.
He e, we should ecall ha (6.1) holds when ε=−1, since (1,b,c)∈R. Mo eo e , aking
in o accoun ha >0i ε=1and ∈(0,1)i ε=−1, we deduce ha , ac ually,
=εH−c2μ+(H−c2μ)2+ε.(7.6)
We now compu e he alue o =(1,b,c)using (4.16). Fi s no e ha , by (7.3), (7.5)
and X( ) =1/c,weha e
||γ|| = 1+ε 2
4cμ.
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 25 o 44 37
Thus using ha κγ=−1/ and (3.26), we ha e om (4.16)
(1,b,c)=−(1+ε 2)
2cμ 1
c+bc1
c+c
b.
Using (7.6) in his exp ession, we ob ain (7.1). 
Rema k 7.2 Fo any p0=(1,b,c)∈R∩O−,c>1, a s aigh o wa d compu a ion
om (7.1) shows ha c(p0)= 0. By he implici unc ion heo em and he analy ici y o
(a,b,c)(P oposi ion 4.9), i ollows ha he le el se (a,b,c)=0,whe e0:= (p0)
can be locally exp essed a ound p0as a g aph c=c0(a,b),whe ecis eal analy ic wi h
espec o a,b.
Simila ly, i p0∈R∩O−,b>1, an analogous compu a ion shows ha b(p0)= 0,
and so in his case we can exp ess locally he le el se (a,b,c)=(p0)as an analy ic
g aph b=b0(a,c).
We now conside ( 1, 3)gi enby(6.4), (6.5). The map (b,c)→ ( 1, 3)is a home-
mo phism om {(b,c):b≥1,c≥1}on o {( 1, 3): 1≤0, 3≥1}, and so we can
iew (1,b,c)as a map ( 1, 3).Using(6.4), (6.5)andμ2=H2+ε, he exp ession o
=( 1, 3)in (7.1) simpli ies o
2=μ(2 3−1)−H
2μ( 3− 1).(7.7)
This can be ew i en al e na i ely as
3=2
2−1 1+H+μ
2μ(1−2)(7.8)
Fo any ixed =0, his equa ion ep esen s he line wi h slope 2
0/(2
0−1) ha passes
h ough he poin
p0:= H+μ
2μ,H+μ
2μ.(7.9)
As an immedia e consequence o (7.1)and(7.7), we ha e:
Co olla y 7.3 (1,b,c)∈(−1,0) o any (1,b,c)∈R∩O−.
We nex show ha in Theo em 7.1 we can simply assume (1,b,c)∈R.
P oposi ion 7.4 R⊂O−.
P oo Le (1,b,c)∈R. Then, (1,b,c)is a o a ional su ace in M3(ε) whose o a ion
axis is o hogonal o e2. A e a linea isome y o R4
ε ha ixes e2,e3, we can assume ha
his o a ion axis is M3(ε) ∩{x1=x2=0}.Wenowconside ,aswedidinSec .4.4, he
s e eog aphic p ojec ion ϕo M3(ε) om −e4 o R3, and he plana cu e γ( ):= ϕ(( )).
De ine nex he numbe

θ:= 1
πσ
0
κγ||γ||d ,
jus as in (4.16). We use he e he new no a ion 
θ,sincein (4.16) was only de ined on O−;
none heless, he igh hand side o (4.16) makes sense in ou case. Mo eo e , he compu a ions
in Theo em 7.1 show ha 
θis also gi en by he igh hand side o (7.1).
123
37 Page 32 o 44 A. Ce ezo e al.
Fig. 3 Le el lines ϒo he pe iod map on he pa ame e domain ( 1, 3) o he cases ε=1 (le ) and ε=−1
( igh ). I 8H2−ε>0, he e a e in ini ely many le el lines o which a segmen o ϒsa is ies he hypo heses
o Theo em 8.4
Fi s , i is clea om (9.2)and0∈J ha ϒ(0)=(0, 3(0)) ∈
W, i.e. he second
condi ion o Theo em 8.4 holds. Also, ϒ( )∈
W o ≥0 small enough; see Fig. 3.
On he o he hand, ϒ( )/∈
W o big enough; indeed, conside he inequali y o 3in
(8.1). I is clea ha 3( )>−εH+μ
2μ o any ≥0, since ϒ( )has nega i e slope. Howe e ,
we can p o e ha
3( )<G( ):= ( 1( )−1)2
1−2 1( )
o la ge enough. This ollows di ec ly om 2
0<1/3(since0∈J) and he inequali ies
0< 
3( )=2
0
1−2
0
<1
2=lim
→∞G( ).
As a consequence, he e is a i s alue >0 such ha ϒ( )/∈
W, o which he equali y in
(8.4) is sa is ied. Thus, he segmen ϒ( ):[0, ]→R2is in he condi ions o Theo em 8.4,
a e he linea change → / .
Thus, by Theo em 8.4, he e exis s ∗>0 such ha ϒ( ∗)∈
W, wi h h(ϒ( ∗)) =0and
so ha h(ϒ( )) changes sign a ∗.Now,le (1,b∗,c∗)be he poin ela ed o ( 1( ∗), 3( ∗))
by he change (6.4), (6.5). Conside he unc ion
g(a,b):= h(a,b,c0(a,b)), (9.3)
de ined on a neighbo hood o (1,b∗)∈R2,whe ec0(a,b)is he analy ic map de ined
in Rema k 7.2, which pa ame izes he le el se (a,b,c)=0in a neighbou hood o
(1,b∗,c∗)∈R3. We know by ou p e ious discussion ha g(1,b∗)=0 and, mo eo e , ha
gchanges sign a b∗, meaning ha g(1,b)<0 ( esp. g(1,b)>0) o b∈(b∗−, b∗)
( esp. b∈(b∗,b∗+)), o >0 small enough. Since g(a,b)is analy ic, he e exis s a
eal analy ic cu e ζ(η) := (a(η), b(η)),η∈[0,δ) o δsmall, sa is ying g(ζ(η)) ≡0and
ζ(0)=(1,b∗). This is a consequence o he classical ac ha i a non-cons an eal analy ic
unc ion :U⊂R2→Rhas a non-isola ed ze o a p0∈U, hen he se { =0}is,
a ound p0, he union o a ini e numbe o eal analy ic a cs mee ing a p0; see e.g. [19,Thm.
5.2.3]. Mo eo e , as b→ g(1,b)changes sign, hen a(η) > 1 o allη>0. We de ine hen
he cu e
C(η) := (a(η), b(η), c0(ζ(η))), (9.4)
123

F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 33 o 44 37
Fig. 4 The le el line ϒand he
line Lmee a a poin ϒ( )∈
W
which by de ini ion is con ained in he le el se (a,b,c)=0, and sa is ies h(C(μ)) ≡0.
Obse e ha we ob ain a di e en cu e C(η) o e e y 0in he coun able se J∩Q.This
concludes he p oo o he i s s ep in he case 8H2−ε>0.
Fi s s ep, case 8H2−ε≤0. This implies in pa icula ha ε=1, and so μ>H.Now,
ake any l>0 such ha l<μ−H
μ+H, and conside he line L( ):= (− ,l +1). Also, o any
0∈−μ−H
2μ,−l
1+l(9.5)
we conside he le el line o he pe iod map ϒ=ϒ( ;0)in (9.2). The condi ions on 0
in (9.5) imply ha ϒand Lmee a a poin L( )=ϒ( ), = (0)>0; see Fig.4.The
alue L( )depends analy ically on 0, and o he limi case 0=−
μ−H
2μi holds =0,
ha is, Land ϒmee a ( 1, 3)=(0,1).
No e also ha , by Lemma 8.5, he inequali y τ(L( )) > u(l +1)holds o all >0
su icien ly close o ze o. In pa icula , he e exis s a smalle in e al
J:= −μ−H
2μ,

such ha , o all 0∈J:
(1) The poin L( )belongs o 
W,andsoτ(L( )) is well de ined.
(2) The inequali y τ(L( )) > u(l +1)is sa is ied.
We now ix some 0∈J∩Qand de ine τ( ):= τ(ϒ( )),h( ):= h(ϒ( )). We will
p o e ha he e exis s a alue ∗∈(0, )whe e τ( ∗)=u( 3( ∗)), wi h 3( )as in (9.2),
and so h( ∗)=0.
By hypo hesis, we know ha ϒ( )=L( )∈
W, and in ac τ( )>u( 3( )), by i em
(2) abo e. Since 3( )is s ic ly inc easing wi h 3(0)<1and 3( )>1, we can de ine
b:= −1
3(1)∈(0, ). Mo eo e , ϒ( b)=(− b,1)does no lie in 
W, since he inequali y
o 3in (8.1) does no hold a ha poin .
Consequen ly, he e exis s a ce ain in e al ( c, ]( b, ]such ha ϒ( )∈
W o all
∈( c, ],andϒ( c)∈∂
W;seeFig.4. By a simila a gumen o he one in he p oo o
Theo em 8.4, i is possible o check ha lim → +
cτ( )=0. On he o he hand, u( 3( )) is a
123
37 Page 34 o 44 A. Ce ezo e al.
posi i e unc ion de ined a = c,so
lim
→ +
c
(τ( )−u( 3( ))) =−u( 3( c)) < 0.
In pa icula , he e exis s some ∗∈( c, )whe e τ( ∗)=u( 3( ∗)). In ac , sinceuand τa e
analy ic and hey do no coincide, we can ake ∗so ha he unc ion ( ):= τ( )−u(u3( ))
changes sign a = ∗, being nega i e ( esp. posi i e) o any < ∗( esp. > ∗)close
enough o ∗.
F om he de ini ion o ∗and P oposi ion 8.3 i ollows ha h( ∗)=0. In ac , since ( )
changes sign a ∗, a guing as in he las pa o he p oo o Theo em 8.4, we deduce ha h( )
also changes sign. Now, le (1,b∗,c∗)∈O−be he poin associa ed wi h ( 1( ∗), 3( ∗)) by
(6.4), (6.5). We deduce ha he analy ic unc ion b→ g(1,b), wi h g(a,b)gi en by (9.3),
changes sign a b=b∗. Consequen ly, as explained in he p oo o he case 8H2−ε>0
abo e, i ollows om he local desc ip ion o he ze o se o eal analy ic unc ions ([19,
Theo em 5.2.3]) ha he e exis s an analy ic cu e ζ(η) =(a(η), b(η)) such ha g(ζ(η)) ≡0,
ζ(0)=(1,b∗)and a(η) > 1 o allη>0. We deduce ha he cu e C(η) de ined as in (9.4),
con ained in he le el se (a,b,c)=0, sa is ies h(C(μ)) ≡0. We ema k ha we ob ain
(a leas ) one such cu e o e e y 0in he coun able se J∩Q. This comple es he i s
s ep o he p oo .
To sum up: so a , we ha e p o ed ha o any alues H≥0, ε=±1 wi h H2+ε>0,
he e is a coun able numbe o cu es Cq:= Cq(η) :[0,δ(q)) →W∩O−, each o hem
con ained in he le el se o he Pe iod map (a,b,c)=q∈J∩Q⊂(−1,0)∩Q, wi h
he p ope y ha h anishes iden ically along Cq(η). Le us now de ine Aq=Aq(η) as
he compac annulus 0=0(a,b,c,τ(a,b,c)) o Theo em 4.12 associa ed o he poin
(a,b,c)=Cq(η). Ou goal is o p o e ha Aq(η) sa is ies each o he p ope ies lis ed in
Theo em 1.1.
I ems (2) and (5) o Theo em 1.1 a e a di ec consequence o Theo em 4.12 and he ac ha
a(η) > 1 o allη>0. Simila ly, i em (3) ollows om P oposi ion 4.10 and i em (4) om
P oposi ion 6.3 and Rema k 8.2. I em (6) holds by cons uc ion, as any su ace =(a,b,c)
has cons an mean cu a u e Hand is olia ed by sphe ical cu a u e lines. Le us now p o e
i em (1). Since h anishes along he cu e Cq(η), we deduce by Co olla y 4.14 ha he annuli
Aq(η) mee o hogonally a ce ain o ally umbilic 2-sphe e Q=Q(q,η) o M3(ε) along
hei bounda y. Le us deno e by B(q,η) hegeodesicballo M3(ε) whose bounda y is he
sphe e Q(q,η); in he case ε=1 he e a e wo such balls, and we choose he one o which
ψ(0,0)∈B(q,η).Fo η=0, we know ha τ(Cq(0)) =u(Cq(0)), so he compac nodoid o
ca enoid Aq(0)is one o he examples cons uc ed in i em 1 o P oposi ion 5.5. In pa icula ,
his o a ional annulus is con ained in B(q,0). By eal analy ici y, we conclude ha he
(non- o a ional) annuli Aq(η),η∈(0,δ
0(q)), will also be con ained in hei espec i e balls
B(q,η), a leas o some δ0(q)>0 small enough. This comple es he p oo o Theo em 1.1.
9.2 P oo o Theo em 1.2
The key idea is o s udy he le el se s o he o m (a,b,c)=−1/n,whe en∈N,n≥2.
Suppose ha ei he ε=−1o H≥1
√3. Then, he inequali y
μ−H
2μ≤1
n2(9.6)
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 35 o 44 37
is sa is ied o some n≥2. Mo e speci ically, i ε=1andH≥1
√3, hen i holds o a ini e
se o na u al numbe s, while i ε=−1i is ue o alln. We will spli ou analysis in o wo
cases, depending on whe he he inequali y (9.6) is s ic o no .
Suppose i s ha (9.6) is s ic o some n≥2. This means ha we a e in he case
8H2−ε>0 de ailed in he p oo o Theo em 1.1,and ha qn:= −1/nlies in he open
in e al Jde ined in (9.1). By he p oo o Theo em 1.1, we deduce ha he e is a cu e
Cqn(η) :[0,δ(n)) →O−, con ained in he le el se (a,b,c)=qn, such ha he associa ed
annuli Aqn(η) a e ee bounda y in a geodesic ball B(qn,η)o M3(ε). I jus emains o check
he embeddedness o hese examples, which will be s udied la e .
Suppose now ha (9.6) holds o he equali y case. In ha si ua ion, we canno use Theo-
em 1.1 di ec ly. By he equali y in (9.6), and using (9.2), he le el cu e (1,b,c)=qn:=
−1/nis exp essed in e ms o ( 1, 3)as
ϒ( ):= ( 1( ), 3( )) =− ,
n2−1+1.
This is he equa ion o a line L( ):= ϒ( ) ha is in he condi ions o Lemma 8.5,so o any
>0 small enough, we ha e ϒ( )∈
Wand τ(ϒ( )) > u( 3( )). Howe e , i is possible
o check ha ϒ( )/∈
W o >0 la ge enough, by a simila a gumen o he i s s ep o
he p oo o Theo em 1.1. Also ollowing he p oo o ha i s s ep, we can deduce ha
he e exis s a poin ∗whe e u( 3( ∗)) =τ(ϒ( ∗)),andsoh(ϒ( ∗)) =0. In ac , we can
show ha he e is a eal analy ic cu e Cn(η) such ha h anishes iden ically along Cn,and
om his we ob ain a 1-pa ame e amily o ee bounda y annuli An(η) ha sa is ies he
p ope ies s a ed in Theo em 1.1.
In conclusion, o any n≥2 such ha (9.6) holds, he e exis s a 1-pa ame e amily
o imme sed, ee bounda y annuli Aqn(η). I jus emains o p o e ha he annuli Aqn(η),
η∈[0,δ(n)), a e embedded o some δ(n)>0 small enough.
Fo e e y η∈[0,δ(n)), le us deno e by ψη(u, )ou usual pa ame iza ion by cu a u e
lines o he compac annulus Aqn(η). We know by Co olla y 4.11 ha ψη(u, )=ψη(u, +
2nσ),whe eσdepends analy ically on η. Iden i ying (u, ) ∼(u, +2nσ) as usual (see
Sec .4.5), we can iew ψηas a pa ame iza ion ψη:[−τ(η),τ(η)]×S1→Aqn(η) o
Aqn(η). Obse e ha o η=0, he annulus Aqn(0)is an embedding, since i is a i ial
co e ing o a c i ical ca enoid o nodoid Nembedded in M3(ε); see Rema k 8.2. Thus, ψ0
is injec i e.
Now, by he eal analy ici y o he amily o compac annuli Aqn(η), we deduce ha o
all ηsu icien ly close o ze o, he pa ame iza ions ψηa e also injec i e, and so he annuli
Aqn(η) a e embedded. This comple es he p oo .
10 Embedded capilla y minimal and CMC annuli in S3
In Theo em 1.2, we cons uc ed embedded examples o non- o a ional, ee bounda y CMC
annuli in geodesic balls o H3( o H>1) and S3( o H≥1/√3). In his sec ion we will
show ha i we elax he ee bounda y condi ion o capilla i y, hen he e exis embedded
non- o a ional capilla y CMC annuli in S3 o any H≥0.
Recall ha a compac su ace in M3(ε) is called a capilla y su ace in a geodesic ball
B⊂M3(ε) i ⊂Bin e sec s ∂Ba a cons an angle along ∂. We p o e nex :
123
37 Page 36 o 44 A. Ce ezo e al.
Theo em 10.1 Fo any H ≥0and any n ≥2 he e exis s a eal analy ic 2-pa ame e amily
o embedded capilla y annuli An(a,η)wi h cons an mean cu a u e H in a geodesic ball
B=B(n,a,η)o S3, wi h a p isma ic symme y g oup o o de 4n.
The ough idea behind his esul is as ollows: conside he le el se o he pe iod
(a,b,c)=−1/n, and suppose ha o some (a,b,c)∈O−wi h a>1in hisle el
se he e exis s a alue u∗>0 such ha m3(u∗)=0, ha is, he hi d coo dina e o he
cen e unc ion anishes. Acco ding o i ems (2), (4) and (5) o Theo em 4.12, he compac
annulus 0(a,b,c,u∗)in e sec s along ∂0a a cons an angle a o ally umbilic sphe e Q
o S3,and0has a p isma ic symme y g oup o o de 4n. Consequen ly, i su ices o check
ha he annuli 0a e embedded and con ained in a geodesic ball Bo S3whose bounda y
is Q.
We will make use o he ollowing lemma:
Lemma 10.2 Suppose ha o some (a0,b0,c0)∈O− he e exis s u0>0such ha
m3(u0)=0. Then, he e exis s a neighbou hood Vo (a0,b0,c0)and an analy ic unc-
ion u∗=u∗(a,b,c):V∩O−→Rsuch ha u∗(a0,b0,c0)=u0and m3(u∗(a,b,c)) ≡0.
P oo The Lemma is a di ec consequence o he implici unc ion heo em i we p o e ha
m
3(u0)= 0. Assume by con adic ion ha m
3(u0)=0. Since m3(u0)=0, we deduce
ha he unc ion m3(u)in (4.7) sa is ies m3(u0)=m
3(u0)=0, due o (4.8). Mo eo e ,
m3sa is ies he di e en ial Eq. (4.9), and so we conclude ha m3(u), and hence, m3(u),
anish iden ically. This is a con adic ion, as (2.10) and he ini ial condi ions (4.11)imply
ha m3(0)=1. 
10.1 P oo o Theo em 10.1
Le n≥2andε=1. We will dis inguish wo cases, depending on whe he o no (9.6) holds.
Assume i s ha (9.6) holds, and conside he le el se (a,b,c)=−1/n=: 0.
Following he p oo o Theo em 1.2 in Sec . 9.2, we know ha he e is a poin (1,b∗,c∗)in
ha le el se such ha h(1,b∗,c∗)=m3(τ(1,b∗,c∗)) =0. By applying Lemma 10.2 wi h
u0=τ(1,b∗,c∗), we deduce he exis ence o a unc ion u∗de ined on a neighbou hood
Vo (1,b∗,c∗)such ha m3(u∗(a,b,c)) anishes iden ically. Now, conside he analy ic
unc ion
(a,b)→ u∗(a,b):= u∗(a,b,c0(a,b)),
whe e c0(a,b)is he analy ic map in Rema k 7.2.Le An(a,b)≡ψ([−u∗,u∗]×S1)be
he compac annulus associa ed o he pa ame e s (a,b,c0(a,b)), whe e we iden i y he
poin s (u, )∼(u, +2nσ)as in Rema k 4.11. By cons uc ion, any o hese annuli mee s a
o ally umbilic sphe e Q(n,a,b)o S3wi h cons an angle along i s bounda y ∂An(a,b),so
in o de o p o e Theo em 10.1 we jus need o check ha he annuli An(a,b)a e embedded
and con ained in a geodesic ball o S3bounded by hei co esponding sphe e Q. No ice ha
in Theo em 1.2 we al eady p o ed his o he annulus An(1,b∗), so by eal analy ici y, he e
is an open neighbou hood G⊂R2o (1,b∗)such ha he annuli An(a,b)a e also embedded
and con ained in a geodesic ball B(n,a,b)o S3bounded by Q(n,a,b) o all (a,b)∈G.
We also ecall ha he e a e wo such geodesic balls in S3; we make he same choice o i
ha we did when p o ing Theo em 1.2.
Take nex some n∈N,n≥2, such ha (9.6) does no hold, and conside he le el cu e
(1,b,c)=0:= −1/n.In he( 1, 3)-coo dina es, his cu e is gi en by ϒ( )in (9.2),
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 37 o 44 37
and so i mee s he ho izon al line 3=1 a a ce ain poin ϒ( α)=(− α,1),whe e α>0.
Le us see ha (1,bα,cα)≡ϒ( α)is in he condi ions o Lemma 10.2.
Le deno e he o a ional H-su ace associa ed o ( 1, 3)=ϒ( α)≡(1,bα,cα); no e
ha cα=1since 3=1, see (6.5). Thus, by Rema k 6.1,co e s a la CMC o us in
S3.Since=−1/n, we can conside o any u0>0 he compac imme sed H-annulus
0=0(1,bα,1,u0)as de ined in Sec .4, a e he iden i ica ion (u, ) ∼(u, +2nσ).
Unde his iden i ica ion, he -cu es ψ(u0, ):S1→S3a e injec i e pa ame iza ions o
ci cles.
We ind nex an explici pa ame iza ion o he p o ile cu e ψ(u,0)o 0,using he
exp esion o he la H- o us in S3gi en in Rema k 5.6 in e ms o he pa ame e s (s,θ).
To s a , no e ha i holds eρ(u)≡1 o 0due o c=1, and so he epa ame izacion
u=u(s)in (8.2)isjus u=2μs. In addi ion, he o a ion axis o he la o us in Rema k 5.6
is S3∩{x1=x2=0}, which ag ees wi h he geodesic o S3 ha con ains he cen e s m(u)
o ; see Rema k 6.4. Thus, by (5.11), we see ha he p o ile cu e o is
ψ(u,0)=μ+H
2μ,0,μ−H
2μsin μ+H
2μu,μ−H
2μcos μ+H
2μu.
(10.1)
Le u0∈(0,u),whe eu:= 2μ
μ+Hπ. F om he exp ession o he p o ile cu e ψ(u,0)
and he p e ious discussion, i ollows ha he pa ame iza ion ψ:[−u0,u0]×S1→S3o
0is injec i e, whe e we iden i y (u, )∼(u, +2nσ)as usual. Mo eo e , 0is con ained
in he ball B[e4,x4(u0)].
We now claim ha he e exis s some u0∈(0,u)such ha m3(u0)=0. To p o e his,
no e i s ha (u):= m(u), ψu(u,0)ne e anishes; indeed, by (2.10), i holds (u)=
eρ
2μ|
N|sin θ. Bu now, we ha e |
N|>0 by cons uc ion, and sin θcanno be ze o by (2.9),
as βis bounded since he unc ions (s, )in (3.11) a e; see also (6.3). The e o e, (u)does
no change sign. On he o he hand, obse e ha , by (10.1),
(0)=m(0), ψu(0,0)=m3(0)
2μ,
(u)=m(u), ψu(u,0)=−m3(u)
2μ.
Since he sign o (u)is cons an in (0,u), we deduce ha he e is some u0∈(0,u) o
which m3(u0)=0. So, by Lemma 10.2, he e exis s a unc ion u∗(a,b,c)de ined on a
neighbou hood o (1,bα,1), wi h u∗(1,bα,1)=u0and m3(u∗(a,b,c)) ≡0.
We conside he analy ic unc ion u∗(a,c):= u∗(a,b0(a,c), c)and de ine o any (a,c)
in a neighbo hood o (1,1) he compac H-annuli
An(a,c):= 0(a,b0(a,c), c,u∗(a,c)),
whe e b0(a,c)is he analy ic map in Rema k 7.2. By cons uc ion, An(a,c)mee s a o ally
umbilic sphe e Q(n,a,c)wi h cons an angle along ∂An(a,c), acco ding o Theo em 4.12.
The annulus An(1,1)is equal o 0(1,bα,1,u0), which by ou p e ious discussion, is
embedded and con ained in a geodesic ball o S3bounded by Q(n,1,1). Consequen ly, he e
is a neighbou hood o (1,1)such ha he annuli An(a,c)a e embedded capilla y CMC
annuli in a geodesic ball o S3bounded by Q(n,a,c). This comple es he p oo o Theo em
10.1.
123

37 Page 38 o 44 A. Ce ezo e al.
A Appendix: P oo o P oposi ion 5.5
In his Appendix we will p o e sepa a ely he h ee i ems o P oposi ion 5.5.
A.1 P oo o i em 1 o P oposi ion 5.5
By he symme ies o S0, i his annulus we e ee bounda y, i should necessa ily be in he
ball B[e4,εx4(

s)]. In such case, S0and S[e4,εx4(

s)]would mee o hogonally along ∂S0.
This happens i and only i he geodesic 
spasses h ough he poin e4, i.e., he cen e o he
ball. As we will see in Appendix A.3, his p ope y holds o he i s posi i e oo 
so he
unc ion
F(s):= x3(s)x(s)−x
3(s)x(s), (A.1)
whe e x,x3a e de ined in (5.9). Consequen ly, ou goal will be o show he exis ence o such

s=
s(H,δ). We will deal wi h se e al cases depending on he alues ε∈{−1,1},H≥0,
δ>0, bu he gene al s a egy is o p o e ha F(0)<0 and ha he e is some s0>0 wi h
F(s0)>0.
We i s conside he case whe e ε=1andδ< H+μ
2,o ε=−1andH>1. Then,
we showed in Sec .5 ha he unc ion x(s)in (5.3) akes alues on he in e al [xm,xM],
whe e xm<xMa e he wo posi i e oo s o h(x); we ecall he e ou assump ion ha
x(0)=xm>0. In his si ua ion, we de ine s2>0 as he i s posi i e alue o which
x(s2)=xM. Consequen ly, x(s)will be s ic ly inc easing o all s∈[0,s2]. We ema k
ha i x≥0, hen h(x)≥0 i and only i x∈[xm,xM]. Addi ionally, i ε=−1, H≤1,
hen x(s)is unbounded, aking alues on [xm,∞),whe exmis he unique posi i e oo o
h(x). The unc ion x(s)will be s ic ly inc easing o all s≥0. In his case, i x≥0, hen
h(x)≥0 i and only i x≥xm.
Le us now show ha F(0)<0. By cons uc ion, x3(0)=0, x(0)=xm>0and
x(0)=0, so i emains o p o e ha x
3(0)>0. By (5.1), (5.5) o ε=1and(5.2), (5.7) o
ε=−1, his is equi alen o he ac ha δ−Hx2
m>0, ha is, x0:= √δ/H>xm. Assume
by con adic ion ha x0≤xm. This implies ha h(x0)≤0, as h(x)≤0in[0,xm].Now,
a di ec compu a ion shows ha h(x0)=x2
0−εx4
0. This quan i y is clea ly posi i e when
ε=−1 and also when ε=1, since x0≤xm<1. We each a con adic ion, wha p o es
ha F(0)<0.
We a e going o show nex ha he e is some s0>0 such ha F(s0)>0, and so he
exis ence o 
s ollows. We spli ou analysis in o i e di e en scena ios: he i s h ee conce n
he sphe ical case (ε=1), while he ou h and i h co e he hype bolic case (ε=−1).
Case 1: ε=1, δ∈(0,H).
In his case, x0=√δ/Hsa is ies xm<x0<xM, so he unc ion s→ δ−Hx(s)2,
and consequen ly φ(s), changes sign on he in e al (0,s2).Le s1∈(0,s2)be he alue
o which φ(s1)=0, so ha φ(s)is inc easing on [0,s1].I φ(s1)>π/2, hen we de ine
s0<s1as he alue o which φ(s0)=π/2. O he wise, le s0=s1. In any o he cases, we
see ha x(s0), x(s0)>0, x
3(s0)<0andx3(s0)≥0, so necessa ily F(s0)>0. We no e
ha he unc ion x4(s)de ined in (5.9) is s ic ly dec easing and posi i e o all s∈[0,s0).
Case 2: ε=1, δ=H.
A di ec compu a ion using (5.3) shows ha xM=1. This implies ha x3(0)=x3(s2)=
0, and since x
3(0)>0, he e exis s a i s s0∈(0,s2)such ha x
3(s0)=0. x3(s)is
inc easing on [0,s0], so in pa icula x3(s0)>0, and hence F(s0)>0. No e also ha he
unc ion φ(s), which mus be inc easing on [0,s0], sa is ies φ(s0)<π
2: o he wise, he e
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 39 o 44 37
Fig. 5 In eg a ion pa h Cn o
(z)
would be an in e media e alue ssuch ha φ(s)=π
2,andsox
3(s)<0 a ha poin ,
eaching a con adic ion. A consequence o his ac is ha x4(s)is dec easing and posi i e
o all s∈[0,s0).
Case 3: ε=1, δ>H.
In his case, δ−Hx2>H(1−x2)≥0, so φ(s)is inc easing o all s∈R.I wemanage
o p o e ha φ(s2)>π/2, hen he e exis s a unique s0∈(0,s2)such ha φ(s0)=π/2. In
pa icula , x3(s0)>0, x
3(s0)<0, and so F(s0)>0. Addi ionally, we deduce ha x4(s)is
dec easing and posi i e o all s∈[0,s0).
Conside he change o a iables w(s)=x(s)2on he in eg al ha desc ibes φ(s2).
De ining wm=w(0)=x2
m,wM=w(s2)=x2
M,wege using(5.3) ha
φ(s2)=wM
wm
δ−Hw
2√w(1−w)w−w2−(Hw−δ)2dw.
We analyze his in eg al ia esidues. No e ha
z−z2−(Hz −δ)2=−(H2+1)(z−wm)(z−wM)(A.2)
by de ini ion o wm,wM, and conside om he e he me omo phic unc ion
(z):= iδ−Hz
2√H2+1(1−z)√z√z−wm√z−wM
de ined on C ((−∞,0]∪[wm,wM])and wi h a pole a z=1. We will in eg a e along
a sequence o closed pa hs CnshowninFig.5. Each such pa h Cncan be di ided in o i e
pieces: a ci cle a c C(1)
no adius n,acu eC(2)
nenclosing he segmen [wm,wM], ano he
cu e C(3)
na ound he in e al (−∞,0]wi h he same bounda y poin s as C(1)
n,andapai
o segmen s S(1)
n,S(2)
nwhich connec C(2)
n,C(3)
n. The segmen s S(1)
n,S(2)
ncoincide bu ha e
opposi e o ien a ions. We ake C(2)
nand C(3)
nso ha hey con e ge o he in e als [wm,wM]
and (−∞,0] espec i ely as ng ows o in ini y.
By he esidue heo em, and using (A.2),
Cn
(z)dz =2πiRes( ,1)=π.
123
37 Page 40 o 44 A. Ce ezo e al.
A ca e ul analysis o (z)shows ha
lim
n→∞C(1)
n
(z)dz =0,
lim
n→∞C(2)
n
(z)dz =2wM
wm
iδ−Hw
2i√H2+1(1−w)√w√w−wm√wM−wdw=2φ(s2),
lim
n→∞C(3)
n
(z)dz =20
−∞
iδ−Hw
2i3√H2+1(1−w)√−w√wm−w√wM−wdw=−M,
o some posi i e cons an M>0. On he o he hand, since he segmen s S(1)
nand S(2)
nha e
opposi e o ien a ions, we deduce ha
S(1)
n
(z)dz =−S(2)
n
(z)dz,
and so we end up wi h
π=2φ(s2)−M<2φ(s2),
as we wan ed o p o e. F om his, he exis ence o 
sis immedia e. I is also possible o deduce
ha x4(s)is dec easing and posi i e o all s∈[0,s0).
Case 4: ε=−1, H>0.
In his case, we de ine s0as he i s alue o which x(s0)=x0,whe ex0=√δ/H.Le
us p o e ha s0exis s. Fi s , i H>1, hen x(s)oscilla es be ween xmand xM, and he ac
ha h(x0)>0 implies ha x0∈(xm,xM),andsos0exis s indeed. In he case H≤1, i holds
x0≥xm.Since o s≥0 he unc ion x(s)is inc easing and sa is ies lims→∞ x(s)=∞,
we see again ha s0exis s. Now, no ice ha φ(s)is an inc easing unc ion on he in e al
[0,s0],andφ(s0)=0. In pa icula , φ(s0)>0. A di ec compu a ion using (5.2)shows
ha
F(s0)=x(s0)sinh(φ(s0))
x(s0)2+1
>0,
as we wan ed o p o e. I is also immedia e o p o e ha εx4(s)=−x4(s)is dec easing o
all s∈[0,s0].
Case 5: ε=−1, H=0.
In his case, φ(s)is an inc easing unc ion sa is ying lims→∞ φ(s)=: φM<∞( he
in eg al in (5.7) is con e gen ). On he o he hand, x(s)inc easing and unbounded, and
sa is ies (5.3). We deduce ha
lim
s→∞ F(s)=lim
s→∞xsinh φ
√x2+1−δcoshφ
√x2+1=sinh(φM)>0,
In pa icula , i ollows ha F(s) anishes o some 
s>0, as we wan ed o p o e. I also
ollows ha −x4(s)is dec easing o all s∈[0,
s].
In any o he conside ed cases, we deduce he exis ence o a i s alue 
s>0 such ha
F(

s)=0. As commen ed be o e, his means ha S0mee s o hogonally he bounda y sphe e
S[e4,εx4(

s)](see Appendix A.3).
Mo eo e , we also ind ha x(s), x(s), x3(s)>0, F(s)<0 o alls∈(0,
s). In addi ion,
x(

s)>0, which shows by (5.6) ha he p incipal cu a u e κs(s)associa ed o he p o ile
cu e mus be s ic ly dec easing on [0,
s+) o some >0. Ano he consequence o hese
inequali ies is ha x
3(s)>0on(0,
s). Since he unc ion x3(s)is odd, we ob ain ha x3(s)
is injec i e on [−
s,
s]. As a esul , he annulus S0mus be embedded.
123
F ee bounda y CMC annuli in sphe ical and hype bolic balls Page 41 o 44 37
Finally, we will show ha S0is ee bounda y in B. I su ices o check ha S0is con ained
in ha ball. This is a consequence o he al eady p o en mono onici y o εx4(s),since
e4,ψ(s,θ)=εx4(s)≥εx4(

s),
o all s∈[0,
s], which shows ha S0⊂B[e4,εx4(

s)].
A.2 P oo o i em 2 o P oposi ion 5.5
We will i s p o e he anali ici y o 
s. This ac is a consequence o he implici unc ion
heo em applied on he unc ion F=F(s;H,δ) de ined in (A.1). We ema k ha his
unc ion no only depends analy ically on s, bu also on he pa ame e s H,δwhich de ine he
unc ions xand x3. I we di e en ia e Fwi h espec o sa s=
s, we ob ain using F(

s)=0
ha
F=−xxx
3
x−x
3x
x2=−xxx
3
x.(A.3)
Since x(

s), x(

s)>0, we jus need o check ha x
3
x= 0a

s o deduce analy ici y. I
ε=1, hen by (5.3), (5.5) and he ac ha F(

s)=0, we deduce a e a long bu di ec
compu a ion ha
x
3
xs=
s=−cos(φ)√1−x2(δ +Hx2)
x2x2<0,(A.4)
whe e he inal quan i y is nega i e since δ>0, H≥0and0<φ(

s)<π/2 (see Appendix
A.1). Simila ly, i ε=−1, hen by (5.3), (5.7), we ob ain a s=
s ha
x
3
xs=
s=−cosh(φ)√x2+1(δ +Hx2)
x2x2<0.(A.5)
In any o he cases, we deduce ha 
s=
s(H,δ)is analy ic.
We will now p o e ha he ex ension o 
s(H,δ)along he bounda y cu e δ=H+μ
2gi en
by (5.10) is con inuous. We conside he unc ion F(s;H,δ) :→Rin (A.1)de inedon
he se
:= (s,H,δ) :s∈R,H≥0,0<δ≤H+μ
2.
We know ha Fis con inuous on since x,x3and hei de i a i es wi h espec o s
depend con inuously on he pa ame e s H,δ. We now claim ha F(s;H,δ) < 0 o all
0≤s<
s,and ha s→ F(s;H,δ) changes sign a s=
s. This is ue when δ< H+μ
2,
acco ding o he esul s in Appendix A.1 and he ac ha F(

s)>0; see (A.3), (A.4). In
he emaining case δ=H+μ
2, i is possible o compu e explici ly he unc ions x(s),x3(s),
which a e gi en by (5.11). This allows us o p o e he claims on Fin his si ua ion, whe e
we a e de iningsH,H+μ
2as in (5.10).
We now ix some H0≥0, and deno e μ0:= H2
0+1. We need o p o e ha he e exis s
lim
(H,δ)→H0,H0+μ0
2
s(H,δ)=π
2√2μ0(H0+μ0)=:
s0.(A.6)
123