P oceedings o he In e na ional Cong ess o Ma hema icians
Hyde abad, India, 2010
Cons an mean cu a u e su aces in
3-dimensional Thu s on geome ies
Isabel Fe n´andez and Pablo Mi a
Abs ac . This is a su ey on he global heo y o cons an mean cu a u e su aces
in Riemannian homogeneous 3-mani olds. These ambien 3-mani olds include he eigh
canonical Thu s on 3-dimensional geome ies, i.e. R3,H3,S3,H2×R,S2×R, he
Heisenbe g space Nil3, he uni e sal co e o PSL2(R) and he Lie g oup Sol3. We will
ocus on he p oblems o classi ying compac CMC su aces and en i e CMC g aphs in
hese spaces. A collec ion o impo an open p oblems o he heo y is also p esen ed.
Keywo ds. Cons an mean cu a u e su aces, homogeneous spaces, Thu s on geome-
ies, ha monic maps, minimal su aces, en i e g aphs.
1. In oduc ion
Cons an mean cu a u e (CMC) su aces appea as c i ical poin s o a na u al geo-
me ic a ia ional p oblem: o minimize su ace a ea wi h o wi hou a olume cons ain
( he uncons ained case co esponds o ze o mean cu a u e, i.e. o minimal su aces). A
undamen al p oblem o his discipline is he geome ic s udy and classifica ion o CMC
su aces unde global hypo heses like compac ness, comple eness, p ope ness o embed-
dedness. The s udy o his p oblem o CMC su aces in he model spaces R3,S3and
H3has p oduced a e y ich heo y, in which geome ic a gumen s in e ac wi h com-
plex analysis, ha monic maps, in eg able sys ems, maximum p inciples, ellip ic PDEs,
geome ic measu e heo y and so on.
One o he mos ema kable achie emen s o his field in he las decade has been
he ex ension o his classical heo y o he case o CMC su aces in simply connec ed
homogeneous 3-dimensional ambien spaces. Apa om R3,S3and H3, hese spaces a e
he emaining fi e Thu s on 3-dimensional geome ies (i.e. H2×R,S2×R, he Heisenbe g
g oup Nil3, he uni e sal co e ing o PSL2(R) and he Lie g oup Sol3), oge he wi h 3-
dimensional Be ge sphe es and some o he Lie g oups wi h le -in a ian me ics (see
Sec ion 2).
I mus be said he e ha he e is an impo an numbe o con ibu ions ega ding
CMC su aces in gene al Riemannian 3-mani olds (no e en homogeneous), many o which
deal o ins ance wi h isope ime ic ques ions o wi h geome ic consequences de i ed om
he s abili y ope a o associa ed o he second a ia ion o he su ace. The achie emen in
he case o homogeneous ambien 3-spaces has been he cons uc ion o a e y ich global
heo y o CMC su aces, analogous o he case o R3,S3and H3, wi h an emphasis on
he geome ic classifica ion (up o ambien isome ies) o p ope ly imme sed o p ope ly
2Isabel Fe n´andez and Pablo Mi a
embedded CMC su aces. The ac ha he ambien space is homogeneous, i.e. i has
he same local geome y a all poin s, makes his p oblem ex emely na u al.
Ou aim he e is o p esen a su ey on some undamen al aspec s o he global heo y
o CMC su aces in homogeneous 3-mani olds. We do no plan, howe e , o gi e a sys-
ema ic accoun o all impo an esul s o his al eady b oad heo y, bu o discuss some
specific p oblems a he co e o i . Hence, he e will be many impo an esul s omi ed,
and we apologize in ad ance o ha .
In o de o explain he p oblems we shall be dealing wi h, le us dis inguish be ween
compac and non-compac CMC su aces in hese spaces.
In he case o compac CMC su aces, h ee undamen al p oblems a e he Alexand o
p oblem (i.e. o classi y compac embedded CMC su aces), he Hop p oblem (i.e. o
classi y CMC sphe es), and he isope ime ic p oblem ( ecall ha isope ime ic egions
on a Riemannian 3-mani old a e bounded by compac embedded CMC su aces, bu he
con e se is no always ue). By classical esul s, ound sphe es cons i u e he solu ion
o each o hese h ee p oblems in he case o CMC su aces in R3. One o ou main
objec i es will be o explain wha is known (and wha is no known) o hese p oblems
in he b oade con ex o CMC su aces in homogeneous 3-mani olds.
In he case o non-compac CMC su aces, one o he basic p oblems is o s udy he
p ope ly embedded CMC su aces o fini e opology. A classical esul in ha di ec ion
is gi en by Be ns ein’s heo em: planes a e he only en i e minimal g aphs in R3. As in
all Thu s on 3-dimensional geome ies he e is a na u al no ion o en i e g aph, i is an
impo an p oblem o he discipline o sol e he Be ns ein p oblem o CMC g aphs, i.e.
o classi y all en i e CMC g aphs in hese 3-dimensional ambien spaces. This will be ou
o he main objec i e.
The heo y o CMC su aces in Thu s on 3-dimensional geome ies s a ed o de elop
as a consis en unified heo y a e some pionee wo ks by Ha old Rosenbe g, join ly wi h
William H. Meeks [MeRo1, MeRo2, Ros] o he case o minimal su aces in p oduc
spaces, and join ly wi h Uwe Ab esch [AbRo1, AbRo2] o he case o CMC su aces in
homogeneous spaces wi h a 4-dimensional isome y g oup.
On one hand, Meeks and Rosenbe g es ablished many esul s on comple e minimal
su aces in M2×R, wha has guided a la ge numbe o subsequen wo ks in he field. A
ecen majo con ibu ion in his sense is he Collin-Rosenbe g heo em [CoRo] on he
exis ence o ha monic diffeomo phims om Con o he hype bolic plane H2, ob ained by
cons uc ing an en i e minimal g aph o pa abolic con o mal ype in H2×R.
On he o he hand, Ab esch and Rosenbe g disco e ed a holomo phic quad a ic di -
e en ial o CMC su aces in hese homogeneous spaces wi h 4-dimensional isome y
g oup ( he E3(κ, τ) spaces), and sol ed he Hop p oblem o hem. The gene al in eg a-
bili y heo y o CMC su aces in he homogeneous E3(κ, τ) spaces was hen es ablished
by B. Daniel [Dan1]. The disco e y by he au ho s o a ha monic Gauss map in o H2
o H= 1/2 su aces in H2×R u ned in o a se ies o pape s by Daniel, Fe n´andez,
Hauswi h, Mi a, Rosenbe g, Sp uck [FeMi1, Dan2, FeMi2, HRS, DaHa] in which he
Be ns ein p oblem o CMC g aphs o c i ical mean cu a u e (including minimal g aphs
in Heisenbe g space Nil3, see Sec ion 6) was sol ed. Ve y ecen ly, he Hop and Alexan-
d o p oblems o CMC su aces ha e been sol ed by Daniel-Mi a and Meeks [DaMi, Mee]
in he emaining Thu s on 3-dimensional geome y: he Lie g oup Sol3, whose isome y
g oup is only 3-dimensional.
We ha e o ganized his exposi ion as ollows. In Sec ion 2 we will in oduce he
3-dimensional homogeneous ambien spaces. In Sec ion 3 we will p esen he basic in-
eg abili y equa ions by Daniel o CMC su aces in he homogeneous spaces E3(κ, τ),
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 3
oge he wi h he holomo phic Ab esch-Rosenbe g diffe en ial, and wi h some basic defi-
ni ions on s abili y o CMC su aces. In Sec ion 4 we will discuss he Hop , Alexand o
and isope ime ic p oblems in he homogeneous spaces E3(κ, τ). Sec ion 5 will be de o ed
o sol ing he Hop and Alexand o p oblems in he eigh h Thu s on geome y, i.e. he
Lie g oup Sol3. In Sec ion 6 we will p esen he solu ion o he Be ns ein p oblem o
en i e g aphs o c i ical CMC in he homogeneous E3(κ, τ) spaces. Finally, in Sec ion
7 we shall expose he Collin-Rosenbe g heo em on pa abolic en i e minimal g aphs in
H2×R, oge he wi h some de elopmen s on he heo y o comple e minimal su aces o
fini e o al cu a u e in H2×R. Mos sec ions finish wi h a selec ion o impo an open
p oblems. See [Mee, DHM] o mo e open p oblems in he heo y.
A mo e de ailed in oduc ion o he global heo y o CMC su aces in homogeneous
3-spaces can be ound in he Lec u e No es by Daniel, Hauswi h and Mi a [DHM].
The au ho s a e g a e ul o H. Rosenbe g, B. Daniel and J.A. G´al ez o use ul ob-
se a ions abou his manusc ip .
2. Homogeneous 3-spaces and Thu s on geome ies
Homogeneous spaces a e he na u al gene aliza ion o space o ms. By defini ion,
a mani old is said o be homogeneous i he isome y g oup ac s ansi i ely on he
mani old. Roughly speaking, he mani old looks he same a all he poin s, e en hough,
s anding a one poin , he mani old can look diffe en in diffe en di ec ions. In he simply
connec ed case, he classifica ion o he 3-dimensional homogeneous spaces is well-known.
I u ns ou ha any simply connec ed homogeneous 3-space mus ha e isome y g oup
o dimension 6, 4 o 3. The comple e lis o hese spaces is he ollowing (see subsec ions
below o mo e de ails):
•The spaces wi h 6-dimensional isome y g oup a e he space o ms: he Euclidean
space R3, he hype bolic space H3(κ), and he s anda d sphe e S3(κ). Fo simplici y
we will assume ha κ=±1 and w i e H3=H3(−1) and S3=S3(1).
•The spaces wi h 4-dimensional isome y g oup a e fib a ions o e he 2-dimensional
space o ms. They a e he p oduc spaces H2×Rand S2×R, he Be ge sphe es,
he Heisenbe g space Nil3and he uni e sal co e ing o he Lie g oup PSL(2,R).
•The spaces wi h 3-dimensional isome y g oup a e a ce ain class o Lie g oups;
among hem we specially quo e he space Sol3.
These spaces a e closely ela ed wi h Thu s on’s Geome iza ion Conjec u e. This
ecen ly p o ed conjec u e s a es ha any compac o ien able 3-mani old can be cu by
disjoin embedded 2-sphe es o o i in o pieces, each one o hem, a e gluing 2-balls o
solid o i along i s bounda y componen s, admi s a geome ic s uc u e. A 3-mani old
wi hou bounda y is said o admi a geome ic s uc u e i i can be endowed wi h a
comple e locally homogeneous me ic. In his case, by conside ing i s uni e sal co e -
ing we ob ain a comple e simply-connec ed locally homogeneous space and hence, by a
esul o Singe , homogeneous. Thus, a 3-mani old admi ing a geome ic s uc u e can
be ealized as he quo ien o a homogeneous simply connec ed 3-space unde he ac ion
o a subg oup o a Lie g oup ac ing ansi i ely by isome ies. The lis o he maximal
geome ic s uc u es ha gi e compac quo ien s consis s o eigh o he p e iously de-
sc ibed spaces: he h ee space o ms, he wo p oduc spaces, Nil3, he uni e sal co e ing
o PSL(2,R) and Sol3(Be ge sphe es mus be excluded om his lis because hey a e
4Isabel Fe n´andez and Pablo Mi a
no maximal, hei isome y g oup a e con ained in he one o he s anda d sphe e S3).
We e e o [Sco, Bon] o mo e de ails.
2.1. Homogeneous spaces wi h 4-dimensional isome y g oup.
Deno e by M2(κ) he 2-dimensional space o m o cons an cu a u e κ( o example,
M2(κ) = R2,H2,S2 o κ= 0,−1,1 espec i ely). As commen ed abo e, any simply
connec ed homogeneous 3-space wi h 4-dimensional isome y g oup admi s a fib a ion
o e M2(κ), o some κ∈R. Mo eo e , hese spaces can be pa ame e ized in e ms o
he base cu a u e κand he bundle cu a u e τ, ha sa is y κ−4τ26= 0. We will use
he no a ion E3(κ, τ) o hese homogeneous spaces.
1. When τ= 0, we ha e he p oduc spaces M2(κ)×R, i.e. up o scaling, he spaces
S2×Rwhen κ > 0, and H2×Rwhen κ < 0.
2. When τ6= 0 and κ > 0, he co esponding spaces a e he Be ge sphe es, a amily
o 2-pa ame e (1-pa ame e a e a homo he ical change o coo dina es) me ics
on he sphe e, ob ained by de o ming he s anda d me ic in such a way ha he
Hop fib a ion is s ill a Riemannian fib a ion. They can also be seen as he Lie
g oup SU(2) endowed wi h a 1-pa ame e amily o le -in a ian me ics.
3. When τ6= 0 and κ= 0, E3(κ, τ) is he Heisenbe g g oup Nil3, he nilpo en Lie
g oup 



1a b
0 1 c
0 0 1
;a, b, c ∈R

,
endowed wi h a 1-pa ame e amily o le -in a ian me ics, all o hem isome i-
cally equi alen a e a homo he ical change o coo dina es.
4. When τ6= 0 and κ < 0, we ob ain he uni e sal co e ing o he Lie g oup PSL(2,R),
endowed wi h a 2-pa ame e (again 1-pa ame e a e homo he ies) amily o le -
in a ian me ics.
The e exis s a common se ing o all hese spaces. Indeed, label D(ρ) = {(x1, x2)∈
R2;x2
1+x2
2< ρ2}. Then, i κ= 0 ( esp. κ < 0), he space E3(κ, τ) can be iewed as R3
( esp. D2/√−κ×R) endowed wi h he me ic
ds2=λ2(dx2
1+dx2
2) + τλ(x2dx1−x1dx2) + dx32, λ =1
1 + κ
4(x2
1+x2
2).(1)
Also, o κ > 0, (R3, ds2) co esponds o he uni e sal co e o E3(κ, τ) minus one fibe .
In all cases, up o a homo he ical change o coo dina es we can suppose wi hou loss o
gene ali y ha κ−4τ2=±1.
The co esponding Riemannian fib a ion π:E3(κ, τ)→ M2(κ) is gi en he e by he
p ojec ion on he fi s wo coo dina es. The uni a y ec o field
ξ=∂
∂x3
is a Killing field angen o he fibe s o π, and will be e e ed o as he e ical field o
he space E3(κ, τ). I sa isfies he equa ion
b
∇Xξ=τX ×ξ
o all ec o fields Xin E3(κ, τ). He e b
∇is he Le i-Ci i a connec ion, × he c oss
p oduc and τ he bundle cu a u e ( his is basically he defini ion o τ).
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 5
A ema kable diffe ence be ween he spaces E3(κ, τ) is ha hei isome y g oup has
ou connec ed componen s in he case τ= 0, and only wo when τ6= 0. This ollows
om he ac ha any isome y in he p oduc spaces can ei he p ese e o e e se he
o ien a ion o he base and he fibe s independen ly, while in he case τ6= 0 i can only
ei he p ese e o e e se bo h o ien a ions. In pa icula , eflec ions only exis in p oduc
spaces.
Also, when τ6= 0 he spaces E3(κ, τ) a e Lie g oups, and i we se σ:= κ
2τ, an
o hono mal ame o le -in a ian ec o fields (called he canonical ame) is gi en by
E1=λ−1cos(σx3)∂
∂x1
+ sin(σx3)∂
∂x2+τ(x1sin(σx3)−x2cos(σx3)) ∂
∂x3
,
E2=λ−1−sin(σx3)∂
∂x1
+ cos(σx3)∂
∂x2+τ(x1cos(σx3) + x2sin(σx3)) ∂
∂x3
,
E3=ξ=∂
∂x3
.
2.2. Homogeneous spaces wi h 3-dimensional isome y g oup.
O all homogeneous spaces wi h 3-dimensional isome y g oup, Sol3is specially impo -
an , since i is he only Thu s on geome y among hem. We will now desc ibe some
aspec s o his space.
A use ul ep esen a ion o Sol3is he space R3wi h he me ic
ds2=e2x3dx2
1+e−2x3dx2
2+dx2
3,
ha is le -in a ian o he s uc u e o Lie g oup gi en by
(x1, x2, x3)·(y1, y2, y3) = (x1+e−x3y1, x2+ex3y2, x3+y3).
The ollowing ec o fields o m an o hono mal le -in a ian ame
E1=e−x3∂
∂x1
, E2=ex3∂
∂x2
, E3=∂
∂x3
.
The isome ies in Sol3a e gene a ed by he h ee 1-pa ame e g oups o ansla ions
(x1, x2, x3)7→ (x1+c, x2, x3),(x1, x2, x3)7→ (x1, x2+c, x3),
(x1, x2, x3)7→ (e−cx1, ecx2, x3+c),
and by he o ien a ion e e sing isome ies fixing he o igin
(x1, x2, x3)7→ (−x1, x2, x3),(x1, x2, x3)7→ (x2,−x1,−x3).
A ema kable ac is he exis ence o wo canonical olia ions, namely
F1={x1= cons an },F2={x2= cons an },
whose lea es a e o ally geodesic su aces isome ic o he hype bolic plane H2. Reflec ions
ac oss any o hese lea es a e o ien a ion e e sing isome ies o Sol3.

6Isabel Fe n´andez and Pablo Mi a
3. CMC su aces: basic equa ions
In his sec ion we p esen h ee impo an ools o ou s udy. One is he se o
in eg abili y equa ions o CMC su aces in E3(κ, τ) by Daniel [Dan1]. Ano he one he
Ab esch-Rosenbe g diffe en ial, a holomo phic quad a ic diffe en ial geome ically defined
on any CMC su ace in E3(κ, τ). The hi d one is a local isome ic co espondence o
CMC su aces in E3(κ, τ) ia which one can pass om one homogeneous space in o ano he
when s udying CMC su aces [Dan1]. Some no ions abou he s abili y ope a o o CMC
su aces a e also gi en.
3.1. In eg abili y equa ions in E3(κ, τ). I is well known ha he Gauss-
Codazzi equa ions a e he in eg abili y condi ions o su ace heo y in R3,S3and H3. In
o he homogeneous spaces, he si ua ion is mo e complica ed.
Le ψ: Σ →E3(κ, τ) be an isome ic imme sion wi h uni no mal map η, and conside
on Σ he con o mal s uc u e gi en by i s induced me ic ia ψ. Associa ed o a con o mal
pa ame e z=s+i on Σ, we will conside he usual ope a o s ∂z= (∂s−i∂ )/2 and
∂¯z= (∂s+i∂ )/2. Also deno e by ξ he e ical Killing field o E3(κ, τ).
De ini ion 3.1. We call he undamen al da a o ψ he 5- uple (λ|dz|2, u, H, p dz2, A dz)
whe e His he mean cu a u e and
λ= 2hψz, ψ¯zi, u =hN, ξi, p =−hψz, Nzi, A =hξ, ψzi.
The unc ion uis commonly called he angle unc ion o he su ace.
Once he e, a se o necessa y and sufficien condi ions o he in eg abili y o CMC
su aces in E3(κ, τ) can be w i en in e ms o hese undamen al da a. This is a esul
by B. Daniel [Dan1], al hough he o mula ion ha we expose he e (i.e. in e ms o a
con o mal pa ame e on he su ace) comes om [FeMi2].
Theo em 3.2 ([Dan1, FeMi2]). The undamen al da a o an imme sed su ace ψ: Σ →
E3(κ, τ)sa is y he ollowing in eg abili y condi ions:

















(C.1)p¯z=λ
2(Hz+uA(κ−4τ2)).
(C.2)A¯z=uλ
2(H+iτ).
(C.3)uz=−(H−iτ)A−2p
λ¯
A.
(C.4)4|A|2
λ= 1 −u2.
(2)
Con e sely, i Σis simply connec ed, hese equa ions a e also sufficien o he exis-
ence o a su ace ψ: Σ →E3(κ, τ)wi h undamen al da a (λ|dz|2, u, H, p dz2, A dz). This
su ace is unique up o ambien isome ies p ese ing he o ien a ions o base and fibe
o E3(κ, τ).
We see hen ha , in he spaces E3(κ, τ), mo e equa ions apa om he Gauss-
Codazzi ones a e needed, due o he loss o symme ies. As a ma e o ac , (C.1) is
he Codazzi equa ion, while he Gauss equa ion does no appea (i is deduced om he
es ). These new equa ions e idence he special cha ac e o he e ical di ec ion in he
E3(κ, τ) spaces.
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 7
De ini ion 3.3. The Ab esch-Rosenbe g diffe en ial o he imme sion is defined as he
quad a ic diffe en ial on Σgi en by
Qdz2=2(H+iτ)p−(κ−4τ2)A2dz2.
I is hen easy o see by means o (C.2) ha he Codazzi equa ion (C.1) can be
eph ased in e ms o Qas
Q¯z=λHz+ (κ−4τ2)H¯zA2
(H+iτ)2.(3)
Consequen ly, one has he ollowing heo em, which gene alized he classical ac ha he
Hop diffe en ial is holomo phic o CMC su aces in R3,S3and H3.
Theo em 3.4 ([AbRo1, AbRo2]). Qdz2is a holomo phic quad a ic diffe en ial on any
CMC su ace in E3(κ, τ).
This is a c ucial esul o he heo y, since i allows he use o holomo phic unc ions
in he geome ic classifica ion o CMC su aces in E3(κ, τ) (see Sec ion 4 and Sec ion 6,
o ins ance).
An impo an ool in he desc ip ion o CMC su aces in R3,S3and H3is he classi-
cal Lawson co espondence. I es ablishes an isome ic one- o-one local co espondence
be ween CMC su aces in diffe en space o ms ha allows o pass, o ins ance, om
minimal su aces in R3 o H= 1 su aces in H3.
The Lawson co espondence was gene alized by B. Daniel o he con ex o homoge-
neous spaces. Indeed, Daniel disco e ed in [Dan1] an isome ic local co espondence o
CMC su aces in all he homogeneous spaces E3(κ, τ), which can be desc ibed as ollows
in e ms o he undamen al da a defined abo e.
Theo em 3.5 (Sis e co espondence, [Dan1]). Le (λ|dz|2, u, H1, p1dz2, A1dz)be he
undamen al da a o a simply connec ed H1-CMC su ace in E(κ1, τ1), and conside
κ2, τ2, H2∈Rso ha
κ2−4τ2
2=κ1−4τ2
1, H2
2+τ2
2=H2
1+τ2
1.
Then i we se θ∈Rgi en by H2−iτ2=eiθ(H1−iτ1), he undamen al da a gi en by
(λ|dz|2, u, H2, p2dz2=e−iθp1dz2, A2dz =e−iθA1dz) (4)
gi e ise o a (simply connec ed) H2-CMC su ace in E3(κ2, τ2), which is locally isome ic
o he o iginal one.
Two su aces ela ed by he abo e co espondence a e called sis e su aces wi h phase
θ. In pa icula , he co esponding Ab esch-Rosenbe g diffe en ials o sis e su aces a e
ela ed by Q2=e−2iθQ1. As special cases o his co espondence we ob ain he associa e
amily o minimal su aces in M2(κ)×R, and a co espondence be ween minimal su aces
in Nil3and CMC 1
2su aces in H2×R. Gene ically, and up o ambien isome ies and
dila ions, he amily o sis e su aces o a gi en choice o (H, κ, τ) is a con inuous 1-
pa ame e amily.
The e is a na u al no ion o g aph in hese spaces. Since E3(κ, τ) has a canonical
fib a ion o e M2(κ) (see Sec ion 2), we will say ha an imme sed su ace Σ in E3(κ, τ)
8Isabel Fe n´andez and Pablo Mi a
is a (local) g aph i he p ojec ion o he base is a (local) diffeomo phism. The CMC-
equa ion o he g aph o a unc ion u=u(x, y) is he PDE (see [Lee])
2H
δ2=∂
∂x α
ω+∂
∂y β
ω,(5)
whe e
δ= 1 + κ
4(x2+y2), ω =p1 + δ2(x2+y2),
α=ux+τy
δ, β =uy−τx
δ.
Fo ins ance, a g aph u=u(x, y) in Nil3≡E3(0, τ) is minimal i and only i i sa isfies
he ellip ic PDE
(1 + β2)uxx −2αβ uxy + (1 + α2)uyy = 0,(6)
whe e α:= ux+y/2 and β:= uy−x/2.
3.2. S abili y and index o CMC su aces. As i is well known, CMC
su aces in Riemannian 3-mani olds appea as he c i ical poin s o he a ea unc ional
associa ed o a ia ions o he su ace wi h compac suppo and cons an enclosed ol-
ume. Equi alen ly, an imme sed su ace Shas cons an mean cu a u e Hi and only i
i is a c i ical poin o he unc ional A ea −2HVol. The second a ia ion o mula o
his unc ional is gi en by
Q( , ) = −ZS
L( ),
whe e Lis he Jacobi ope a o (o s abili y ope a o ) o he su ace:
L= ∆ + ||B||2+ Ric(η).
He e ∆ is he Laplacian o he induced me ic on he su ace, Bis he second undamen al
o m, ηis he uni no mal ec o field, and Ric is he Ricci cu a u e in he ambien
mani old. As a pa icula case, he Jacobi ope a o o CMC su aces in he spaces
E3(κ, τ) can be ew i en (see [Dan1]) as
L= ∆ −2K+ 4H2+ 4τ2+ (κ−4τ2)(1 + u2),
being K he Gaussian cu a u e o he su ace and u he angle unc ion (see Defini ion
3.1). A Jacobi unc ion is a unc ion o which L( ) = 0.
A CMC su ace Sis said o be s able ( esp. weakly s able) i
Q( , ) = −ZS
L( )≥0
holds o any smoo h unc ion on Swi h compac suppo ( esp. wi h compac suppo
and RS = 0). Fo ins ance, CMC g aphs in E3(κ, τ) a e s able, and compac CMC
su aces bounding isope ime ic egions a e weakly s able (bu no necessa ily s able, as
ound sphe es in R3show).
An impo an concep ela ed o s abili y is he index o a CMC su ace. The index
o a compac CMC su ace is defined as he numbe o nega i e eigen alues o i s Jacobi
ope a o . Thus, s able CMC su aces (in pa icula CMC g aphs) ha e index ze o. Round
sphe es in R3ha e index one.
We e e o [MPR] o mo e de ails abou s abili y o CMC su aces.
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 9
4. Compac CMC su aces in E3(κ, τ)
In his sec ion we explain he mos impo an esul s ha a e known ega ding he
exis ence and uniqueness o compac CMC su aces in he homogeneous 3-spaces E3(κ, τ).
The undamen al examples a e he o a ional CMC sphe es, and we shall be in e es ed in
hei uniqueness among compac embedded CMC su aces, and among imme sed CMC
su aces. These p oblems a e called, espec i ely, he Alexand o and Hop p oblems.
4.1. Ro a ional compac CMC su aces. Al hough ound sphe es in he
model spaces R3,S3,H3a e CMC sphe es, his does no hold o he es o homogeneous
spaces. Howe e , in all he spaces E3(κ, τ) he e exis o a ions wi h espec o he
e ical axis, and so he e is a na u al no ion o o a ional su ace. I is hence na u al
o seek CMC sphe es (and CMC o i) in E3(κ, τ) among he class o o a ional su aces.
This can be done by ODE analysis, and he esul o his can be summa ized as ollows:
Theo em 4.1. (S uc u e o o a ional CMC sphe es in E3(κ, τ)).
1. I κ−4τ2>0, hen o e e y H∈R he e exis s a unique o a ional CMC Hsphe e
(up o isome ies) in E3(κ, τ). These sphe es a e embedded i τ= 0, i.e. in S2×R,
and also o mos Be ge sphe es. Howe e , o some Be ge sphe es wi h small
bundle cu a u e τ(wi h espec o a fixed κ) he e is a ce ain egion o a ia ion
o he pa ame e s (H, τ)whe e he sphe es a e non-embedded. This egion can be
explici ly desc ibed, see [To ].
2. I κ−4τ2<0, hen
•i H2⩽−κ
4, hen he e exis s no o a ional CMC Hsphe e in E3(κ, τ),
•i H2>−κ
4, hen he e exis s a unique o a ional CMC Hsphe e (up o
isome ies) in E3(κ, τ). All hese sphe es a e embedded.
Le us ema k ha all hese CMC sphe es can be cons uc ed explici ly. We shall call
hem canonical o a ional CMC sphe es. Fo example, he o a ional CMC Hsphe es in
S2×R⊂R4a e gi en by he o mula
ψ(u, ) = (−cos k(u),sin k(u) cos , sin k(u) sin , h(u)),
whe e −1≤u≤1, H∈Rand
k(u) := 2 a c an 2H
√1−u2, h(u) := 4H
√4H2+ 1 a csinh u
√1−u2+ 4H2.
Besides hese o a ional CMC sphe es, he e also exis o a ional CMC o i in E3(κ, τ)
when (and only when) κ−4τ2>0 (excluding minimal su aces in S2×R). Fo S2×R,
hey a e all embedded (see Ped osa [Ped]). Fo Be ge sphe es he si ua ion is explained
by To albo and U bano in [To , ToU ]; one has o e e y H o a ional embedded CMC
o i gi en by he Hop li o a ci cle in S2, bu he e also exis some o he non-fla
o a ional CMC o i. The embeddedness p oblem o such o i is open in gene al, bu
o he minimal case he e a e embedded o a ional o i o he han Cliffo d o i. This
con as s wi h he case o embedded minimal o i in S3.
A gene al s udy o CMC su aces in H2×Rand S2×Rin a ian by a con inuous
1-pa ame e subg oup o ambien isome ies can be ound in [SaE, SaTo].
16 Isabel Fe n´andez and Pablo Mi a
De ini ion 6.3. We will say ha a ha monic map Gin o H2admi s Weie s ass da a
{Q0, τ0}i he pullback me ic induced by Gcan be w i en as
hdG, dGi=Q0dz2+µ|dz|2+¯
Q0d¯z2, µ =τ0
4+4|Q0|2
τ0
,
τ0being a posi i e smoo h unc ion.
6.1.1. H= 1/2 su aces in H2×R.We will ega d H2×R=E3(−1,0) in i s
Minkowski model, i.e.
H2×R={(x0, x1, x2, x3) : x0>0,−x2
0+x2
1+x2
2=−1} ⊂ L3×R= L4.
Using his model, he uni no mal ec o ηo an imme sed su ace ψ= (N, h) : Σ →
H2×R akes alues in he de Si e 3-space, and {η, N}is an o hono mal ame o he
Lo en zian no mal bundle o ψin L4. Mo eo e , i uis he angle unc ion o he su ace
( ha is, he las coo dina e o η) and we assume ha u6= 0 ( ha is, ha ψis nowhe e
e ical, o equi alen ly, ha i is a mul ig aph), hen we can w i e
1
u(η+N) = (G, 1),(12)
o a ce ain map G: Σ →H2.
De ini ion 6.4 ([FeMi1]). The map Ggi en by (12) will be called he hype bolic Gauss
map o an imme sed (nowhe e e ical) su ace in H2×R.
The main p ope y o he hype bolic Gauss map is he ollowing [FeMi1]:
Theo em 6.5 (Fe n´andez-Mi a). The hype bolic Gauss map o a CMC su ace wi h
H= 1/2in H2×Ris a ha monic map in o H2, and admi s Weie s ass da a {−Q, λu2},
whe e Qdz2,λ|dz|2and ua e, espec i ely, he Ab esch-Rosenbe g diffe en ial, he me ic,
and he angle unc ion o he su ace.
Con e sely, i Σis simply connec ed, any ha monic map G: Σ →H2admi ing
Weie s ass da a is he hype bolic Gauss map o some H= 1/2su ace in H2×R.
Mo eo e , he space o H= 1/2su aces in H2×Rwi h he same hype bolic Gauss
map Gis gene ically wo-dimensional, and i can be eco e ed om Gby a ep esen a ion
o mula.
The p oo o he di ec pa o he abo e esul ollows om equa ions (2) and he
e y defini ion o G. The con e se pa is an in eg abili y a gumen . This esul is o
g ea impo ance o he es o his sec ion, since i allows he use o ha monic maps in
he desc ip ion o su aces o c i ical CMC.
6.1.2. Minimal su aces in Nil3.The exis ence o his ha monic Gauss map o
H= 1/2 su aces in H2×Rwas ex ended by B. Daniel [Dan2] o he case o minimal
su aces in Nil3=E3(0,1
2).
This ime, he ha monic Gauss map is gi en by he Lie g oup Gauss map o he
su ace. Indeed, i we iden i y he Lie algeb a o Nil3wi h he angen space a a poin
by le mul iplica ion, we can s e eog aphically p ojec he uni no mal ec o field o
ob ain a map aking alues in he ex ended complex plane. Mo e specifically, we will
conside he model o Nil3gi en in Sec ion 2 and i s canonical ame o le -in a ian

Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 17
fields {E1, E2, E3}. I N=PNiEiis he uni no mal o X: Σ →Nil3, hen he Gauss
map o Xis gi en by
g=N1+iN2
1 + N3
: Σ →C.
Now, i he su ace is nowhe e e ical we can o ien i so ha u=hN, E3iis posi i e,
and so g akes alues in he uni disc D. By iden i ying H2wi h (D, ds2
P), whe e ds2
Pis
he Poinca ´e me ic, Daniel ob ained in [Dan2]:
Theo em 6.6 (Daniel). The Gauss map o a nowhe e e ical minimal su ace is ha -
monic in o H2.
Con e sely, le g: Σ →H2be a ha monic map defined on a simply connec ed o ien ed
Riemann su ace in o H2, and assume ha gis nowhe e an iholomo phic (i.e., gzdoes
no anish a any poin ). Take z0∈Σand X0∈Nil3.
Then he e exis s a unique con o mal nowhe e e ical minimal imme sion X: Σ →
Nil3wi h X(z0) = X0ha ing gas i s Gauss map. Mo eo e , Xcan be uniquely eco e ed
om g h ough an adequa e ep esen a ion o mula.
Fu he mo e, i can be checked ha he Weie s ass da a o gas abo e a e {−Q, λu2},
whe e Qdz2,λ|dz|2and ua e, espec i ely, he Ab esch-Rosenbe g diffe en ial, he me ic,
and he angle unc ion o he su ace defined in Sec ion 2.
As we saw in Sec ion 3, minimal su aces in Nil3and H= 1/2 su aces in H2×Ra e
ela ed by he sis e co espondence, and sis e su aces ha e he same me ic and angle
unc ion (in pa icula , he condi ion o being nowhe e e ical is p ese ed). As in his
case he sis e su aces ha e opposi e Ab esch-Rosenbe g diffe en ials, i u ns ou ha
hei espec i e ha monic Gauss maps a e conjuga e o each o he .
The ela ion be ween minimal su aces in Nil3and H= 1/2 su aces in H2×Rcan be
made mo e explici by means o he heo y o spacelike CMC su aces in L3, as ollows.
Theo em 6.7 ([FeMi3]). Le X= (F, ) : Σ →Nil3be a simply connec ed nowhe e
e ical minimal su ace wi h me ic λ|dz|2and angle unc ion u, and ψ= (N, h) : Σ →
H2×Ri s sis e su ace.
Then := (F, h) : Σ → L3is a spacelike H= 1/2su ace in he Minkowski 3-space
wi h me ic λu2|dz|2and Hop diffe en ial −Qdz2, whe e Qdz2is he Ab esch-Rosenbe g
diffe en ial o X.
6.1.3. CMC √−κ/2 su aces in ^
PSL(2,R). In a o hcoming pape [DFM], he
au ho s and B. Daniel will p o e ha he e exis s also a ha monic Gauss map o c i ical
CMC su aces in he emaining case, i.e. he uni e sal co e ing o he g oup PSL(2,R),
and will de i e a ep esen a ion o mula o hem.
6.2. Hal -space heo ems. One o he mos impo an esul s in he global
s udy o minimal su aces in R3is he classical hal -space heo em by Hoffman and Meeks
[HoMe]. This heo em says ha any p ope ly imme sed minimal su ace in R3lying in a
hal -space mus be a plane pa allel o he one de e mining he hal -space. The main ools
used he e a e he maximum p inciple and he exis ence o ca enoids, a 1-pa ame e amily
o minimal su aces con e ging o a doubly-co e ed punc u ed plane P, and in e sec ing
he planes pa allel o Pin compac cu es.
The analogous e sion o CMC one hal su aces in H2×Rwas p o ed in [HRS]. In
his se ing, ho ocylinde s play he ole o he planes in R3.
18 Isabel Fe n´andez and Pablo Mi a
Theo em 6.8 (Hauswi h-Rosenbe g-Sp uck). The only p ope ly imme sed CMC one
hal su aces in H2×R ha a e con ained in he mean con ex side o a ho ocylinde C
a e he ho ocylinde s pa allel o C.
Also, he only p ope ly embedded CMC one hal su aces in H2×Rcon aining a
ho ocylinde in i s mean con ex side a e he ho ocylinde s.
P oo . The main poin he e is o cons uc a amily o CMC one hal su aces in H2×R
o be used in he same way as ca enoids in he p oo o he hal -space heo em in R3.
This is achie ed by means o compac annuli wi h bounda ies, con ained be ween wo
ho ocylinde s.
Fo he case o Nil3, we mus dis inguish be ween ho izon al and e ical hal -spaces.
The equi alen o he hal -space heo em o su aces lying in a ho izon al hal -space
is p o ed by using he amily o o a ional annuli [AbRo2]. The co esponding e ical
e sion has been ob ained in [DaHa], by cons uc ing fi s a amily o ho izon al ca enoids,
i.e. p ope ly embedded minimal annuli (non- o a ional) wi h a geome ic beha iou good
enough o apply he Hoffman-Meeks echnique.
Theo em 6.9 (Daniel-Hauswi h). The only p ope ly imme sed minimal su aces in
Nil3 ha a e con ained in a e ical hal space a e he e ical planes pa allel o he one
de e mining he hal -space.
P oo . Using he ep esen a ion o mula o minimal su aces in Nil3(see Theo em 6.6),
i is possible o cons uc ho izon al ca enoids in Nil3. These su aces a e a 1-pa ame e
amily o p ope ly embedded minimal annuli, in e sec ing e ical planes {x2=c}in a
non-emp y closed con ex cu e. Mo eo e , he amily con e ges o a double co e ing o
{x2= 0}minus a poin . They a e ob ained by in eg a ing a amily o ha monic maps
ha belong o a mo e gene al amily used in he cons uc ion o Riemann ype minimal
su aces in H2×R[Ha]. Once we ha e hese ca enoids, we finish by using he maximum
p inciple simila ly o he Euclidean case.
6.3. The classi ica ion o en i e g aphs. In his sec ion we will desc ibe
he space o en i e g aphs o c i ical CMC in E3(κ, τ). Such a desc ip ion ollows om
he wo ks o Fe n´andez-Mi a [FeMi1, FeMi3], Hauswi h-Rosenbe g-Sp uck [HRS] and
Daniel-Hauswi h [DaHa], and is con ained in Theo ems 6.10 and 6.11. We expose he e
a unified pe spec i e o his subjec . Fi s , we ha e
Theo em 6.10 ([DaHa, FeMi3, HRS]). The ollowing condi ions a e equi alen o a
su ace o c i ical CMC in E3(κ, τ):
(1) I is an en i e g aph.
(2) I is a comple e mul ig aph.
(3) u2ds2is a comple e Riemannian me ic (whe e uis he angle unc ion and ds2 he
me ic o he su ace).
In pa icula , he sis e co espondence p ese es en i e g aphs o c i ical CMC.
Le us make some commen s on his heo em. Fi s , Hauswi h, Rosenbe g and
Sp uck p o ed (2) ⇒(1) o H= 1/2 su aces in H2×R. Second, he au ho s p o ed
in [FeMi3] ha (3) ⇒(1) ( o any su ace in E3(κ, τ), no necessa ily CMC), and ha
(1) ⇒(3) holds o minimal su aces in Nil3. Finally, Daniel and Hauswi h showed ha
(2) ⇒(1) holds o minimal su aces in Nil3. The es o he cases can be easily ob ained
om hese esul s and he sis e co espondence ( his was fi s obse ed in [DHM]).
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 19
P oo . I is immedia e ha (1) ⇒(2). Also, by an eigen alue es ima e, he au ho s
p o ed in [FeMi3] ha o a bi a y su aces in E3(κ, τ) i holds u2ds2≤gF, whe e
F=π◦ψis he p ojec ion on o M2(κ) o ψ. Thus, i u2ds2is comple e, Fis a local
diffeomo phism wi h comple e pullback me ic, and by s anda d opological a gumen s,
Fis a diffeomo phism, i.e. (3) ⇒(1) holds.
Tha (1) ⇒(3) holds o minimal su aces in Nil3was also p o ed in [FeMi3]: le
X= (F, ) : Σ →Nil3be an en i e minimal g aph. By Theo em 6.7, he e is an en i e
spacelike CMC g aph = (F, h) : Σ → L3, whose induced me ic is ds2
=u2ds2. Now
we can apply a heo em by Cheng and Yau [ChYa] which says ha spacelike en i e CMC
g aphs in L3ha e comple e induced me ic. Hence u2ds2is comple e, as wished.
We will now p o e ha (2) ⇒(1) holds o minimal su aces in Nil3. Le us obse e
ha once his is done, we can also p o e he heo em o su aces o c i ical CMC in all
he spaces E3(κ, τ). Indeed, as any simply connec ed su ace o c i ical CMC is he sis e
su ace o some minimal su ace in Nil3, and as he co espondence p ese es he me ic
and he angle unc ion ( he e o e i p ese es condi ions (2) and (3) by passing o he
uni e sal co e ing), we can easily ansla e he heo em o he case o minimal su aces
in Nil3 o he es o he spaces. I is impo an he e ha we p o ed (3) ⇒(1) in all
spaces.
So, we only need o p o e (2) ⇒(1) o minimal su aces in Nil3. This was done
by Daniel and Hauswi h [DaHa]. Fo ha , hey used hei hal -space heo em in Nil3
(Theo em 6.9) and an adap a ion o Nil3o he p e ious p oo o (2) ⇒(1) o he case
o H= 1/2 su aces in H2×Rgi en by Hauswi h-Rosenbe g-Sp uck [HRS].
In o de o p o e (2) ⇒(1) o minimal su aces in Nil3, we a gue by con adic ion.
Assume ha he e exis s a comple e mul ig aph Σ ha is no en i e. Then he e exis s
an open se Σ0⊂Σ ha is a g aph o e a disc D( R2o a unc ion , and a poin
q∈∂D such ha does no ex end o q.
S ep 1: Fo any sequence o poin s {qn}in Dcon e ging o q, he sequence o no mal
ec o s a he poin s pn= (qn, (qn)) ∈Σ0con e ges o he ho izon al ec o o hogonal
o ∂D a q.
Indeed, as he su ace is a mul ig aph, i s angle unc ion u=hN, E3i, whe e N
deno es he uni no mal ec o , is a Jacobi unc ion ha does no anish. As a esul o
his, Σ is (s ongly) s able, and has bounded geome y. This means ha locally a ound
any pnwe can w i e he su ace as he g aph (in exponen ial coo dina es) o e a disc o
adius δo i s angen plane, whe e δis a uni e sal cons an depending only on Σ. This
neighbo hood o pnwill be deno ed by G(pn). The limi o he no mal ec o s {N(pn)}
mus be a ho izon al ec o since o he wise, he piece G(pn) o bounded geome y could
be ex ended as a g aph beyond q, which is impossible. Mo eo e , he limi ec o mus
be no mal o ∂D a qsince Σ0is a g aph o e D.
S ep 2: The unc ion defining he g aph Σ0di e ges a q. Mo eo e , as we app oach
q, and a e ansla ing he su ace o he o igin, he su aces con e ge o a piece o he
( ansla ed) e ical plane Ppassing h ough qand angen o ∂D.
Tha di e ges a qis a consequence o he comple eness o Σ, and he las pa can
be p o ed by ollowing he ideas o Collin and Rosenbe g in [CoRo]. We will assume ha
Pis he plane {x1=c}.
S ep 3: Σcon ains a g aph Go e a domain o he o m Uǫ= (c−ǫ, c)×R⊂R2.
Mo eo e , his g aph is disjoin om Pand asymp o ic o i as one app oaches q.
The g aph Gis ob ained by analy ical con inua ion o he su aces G(pn) used in he
fi s s ep, and a e a ca e ul s udy o he beha io o he in e sec ion cu es o hese
g aphs and he planes pa allel o P.
20 Isabel Fe n´andez and Pablo Mi a
Finally, he con adic ion ollows om he hal -space heo em (Theo em 6.9). Re-
call ha , al hough Ghas bounda y and he heo em is o mula ed o su aces wi hou
bounda y, i s p oo applies o his case, and so we a e done.
Once he e, we in es iga e he Be ns ein p oblem o en i e g aphs o c i ical CMC
in E3(κ, τ), i.e. he classifica ion o such en i e g aphs ( ecall he e ha CMC g aphs in
E3(κ, τ) sa is y he ellip ic PDE (5)). The e minology comes om he classical Be ns ein
heo em: en i e minimal g aphs in R3a e planes. Equi alen ly, any solu ion o he
minimal g aph equa ion
(1 + 2
y) xx −2 x y xy + (1 + 2
x) yy = 0 (13)
defined on he whole plane is linea .
I is in e es ing o compa e his esul wi h he Be ns ein p oblem in Nil3, i.e. he
classifica ion o en i e minimal g aphs in Nil3. This co esponds o classi ying all global
solu ions o he PDE (6). Obse e ha aking τ= 0 in (6) we ob ain he classical
equa ion (13), i.e. he classical case conside ed by Be ns ein appea s as a limi o he
Heisenbe g case.
The e exis s, howe e , a g ea diffe ence be ween bo h si ua ions. The ollowing esul
classifies he en i e g aphs o c i ical CMC in E3(κ, τ), by pa ame izing he moduli space
o such en i e g aphs in e ms o holomo phic quad a ic diffe en ials. I was ob ained fi s
o minimal g aphs in Nil3by he au ho s [FeMi3], and sho ly he ea e by Daniel and
Hauswi h [DaHa] o H= 1/2 g aphs in H2×R. The gene al case ollows easily om he
Heisenbe g case and Theo em 6.10, using he sis e co espondence ( his was obse ed
fi s in [DHM]).
Theo em 6.11 (Fe n´andez-Mi a, Daniel-Hauswi h). Le Qdz2deno e a holomo phic
quad a ic diffe en ial on Σ≡Co D, such ha Q6≡ 0i Σ≡C, and le H2=−κ/4.
The e exis s a 2-pa ame e amily o en i e CMC Hg aphs in E3(κ, τ)whose Ab esch-
Rosenbe g diffe en ial ag ees wi h Qdz2. These g aphs a e gene ically non-cong uen .
And con e sely, hese a e all he en i e g aphs o c i ical CMC in E3(κ, τ).
A his poin , he p oo o he case o minimal su aces in Nil3is a consequence o
Theo em 6.7 and he ollowing esul by Wan and Wan-Au [Wan, WaAu] on spacelike
en i e CMC g aphs in L3: o any holomo phic quad a ic diffe en ial as abo e, he e exis s
a unique (up o isome ies) spacelike en i e CMC 1/2g aph in L3wi h Hop diffe en ial
Qdz2.The 2-pa ame e amily o non-cong uen g aphs in E3(κ, τ) comes om he loss
o ambien isome ies ( om 6 dimensions o 4 dimensions) when passing om L3 o Nil3.
The emaining cases o c i ical CMC g aphs ollow since by Theo em 6.10 he sis e
co espondence p ese es en i e g aphs.
6.4. Open P oblems. As explained in Sec ion 3, en i e g aphs a e s able. I
is conjec u ed ha en i e g aphs and e ical cylinde s a e he only s able c i ical CMC
su aces ( his has been p o ed o pa abolic con o mal ype in [MaPR]). Rela ed o his
is he ques ion o non-exis ence o comple e s able H > 1/2 su aces in H2×R(p o ed
o H > 1/√3 by Nelli-Rosenbe g, [NeRo2]).
Also, no much is known abou p ope ly embedded su aces o c i ical CMC and
non- i ial opology. Can one ob ain hem by conjuga e Pla eau cons uc ions, o by
in eg able sys ems echniques? Ano he ema kable p oblem is o es ablish he s ong
hal -space heo em in Nil3: a e wo disjoin p ope ly embedded minimal su aces in Nil3
necessa ily wo pa allel e ical planes, o wo pa allel en i e minimal g aphs?
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 21
7. Minimal su aces in H2×Rand S2×R
Minimal su aces in p oduc spaces admi a special ea men , due o se e al easons.
One o hem is he ollowing: i ψ= (N, h) : Σ →M2×Ris a minimal su ace imme sed
in he p oduc space M2×R, whe e (M2, g) is a Riemannian su ace, hen he ho izon al
p ojec ion N: Σ →M2is a ha monic map and he heigh unc ion h: Σ →Ris a
ha monic unc ion. This implies, o ins ance, ha compac minimal su aces in M2×R
only exis i M2is compac (in pa icula , i M2=S2), and he only ones a e he slices
M2×{ 0}.
Ano he impo an ac abou minimal su aces in M2×Ris ha he e is a na u al
no ion o minimal g aph o e a domain Ω ⊂M2, and ha his g aph sa isfies a simple
ellip ic PDE in di e gence o m. This ac oge he wi h gene al exis ence esul s o
solu ions o he Pla eau p oblem in Riemannian 3-mani olds allows a good con ol on he
geome y o he su ace. Some o he mos in e es ing esul s o he heo y o minimal
su aces in p oduc spaces come om he in e play be ween he in o ma ion p o ided by
ha monic maps and by Pla eau cons uc ions and he minimal g aph equa ion.
S a ing wi h he pionee wo k o H. Rosenbe g [Ros], and W.H Meeks and H. Rosen-
be g [MeRo1, MeRo2], he heo y o minimal su aces in M2×Rhas de eloped subs an-
ially in he las decade. We will only alk he e abou a ew esul s o special ele ance
o he heo y, and no men ion many o he impo an esul s.
7.1. The Collin-Rosenbe g heo em. The classical Be ns ein heo em in
R3s a es ha planes a e he only en i e minimal g aphs in R3. This heo em can be
ex ended o he case o p oduc spaces: any en i e minimal g aph in M2×R, whe e
(M2, g)is a comple e su ace o non-nega i e cu a u e, is o ally geodesic.
In con as , in he p oduc space H2×R he e is a wide a ie y o en i e minimal
g aphs. Fo ins ance, in [NeRo1] Nelli and Rosenbe g sol ed he Di ichle p oblem a
infini y o he minimal g aph equa ion in H2×R. They p o ed ha any Jo dan cu e
a he ideal bounda y S1×R≡∂∞H2×Ro H2×Rwhich is a g aph o e S1≡∂∞H2is
he asymp o ic bounda y o a unique en i e minimal g aph in H2×R(see [GaRo] o a
p oo o his in he mo e gene al case o en i e minimal g aphs in M2×R, whe e (M2, g)
is comple e, simply connec ed and wi h KM≤c < 0).
All hese en i e minimal g aphs a e hype bolic, ha is, hey ha e he con o mal ype
o he uni disk. The p oblem o exis ence o en i e minimal g aphs o pa abolic ype
(i.e. wi h he con o mal ype o C) is much ha de , and was sol ed ecen ly by Collin and
Rosenbe g [CoRo].
Theo em 7.1 (Collin-Rosenbe g). The e exis en i e minimal g aphs in H2×Ro
pa abolic con o mal ype.
As he p ojec ion on o H2o a minimal g aph is a ha monic diffeomo phism, he abo e
heo em has he ollowing consequence, which sol es a majo p oblem in he heo y o
ha monic maps and disp o es a conjec u e by R. Schoen and S.T. Yau.
Co olla y 7.2 (Collin-Rosenbe g). The e exis ha monic diffeomo phisms om Con o
H2.
The p oo by Collin and Rosenbe g is a good example o he in e ac ion be ween he
ha monici y p ope ies o he minimal imme sion and he use o Pla eau cons uc ions
and he minimal g aph equa ion.

22 Isabel Fe n´andez and Pablo Mi a
The main idea in he p oo is o cons uc fi s (non-en i e) minimal g aphs in H2×R
o Sche k ype o e ideal geodesic polygons, ha ing al e na ing asymp o ic alues +∞
and −∞ on he sides o he polygon. This gene alizes a classical cons uc ion by Jenkins
and Se in in he case o minimal g aphs o e bounded domains in R3. This cons uc ion
is done as ollows:
Le Γ be an ideal polygon o H2, so ha all he e ices o Γ a e a he ideal bounda y
o H2and Γ has an e en numbe o sides A1, B1, A2, B2..., Ak, Bk, o de ed clockwise. A
each e ex ai, we conside a small enough ho ocycle Hiwi h Hi∩Hj=∅. Each Ai
( esp. Bi) mee s exac ly wo ho ocycles. Deno e by e
Ai( esp. e
Bi), he compac a c o
Ai( esp Bi) which is he pa o Aiou side he wo ho odisks. We deno e by |Ai| he
leng h o |e
Ai|. Define e
Biand |Bi|in he same way.
Now we can conside a(Γ) = Pk
i=1 |Ai|and b(Γ) = Pk
i=1 |Bi|. We obse e ha
a(Γ) −b(Γ) does no depend on he choice o he ho ocycle Hia ai, since ho ocycles
wi h he same poin a infini y a e equidis an . Keeping in mind hese da a, we can s a e
he ollowing heo em by Collin-Rosenbe g [CoRo] (see also Nelli-Rosenbe g [NeRo1]):
Theo em 7.3. ([NeRo1], [CoRo]) The e is a (unique up o addi i e cons an s) solu ion
o he minimal su ace equa ion in he polygonal domain P, equal o +∞on Aiand −∞
on Bi, i and only i he ollowing condi ions a e sa isfied:
1. a(Γ) = b(Γ),
2. Fo each insc ibed polygon Pin Γ,P 6= Γ, and o some choice o ho ocycles a he
e ices, one has
2a(P)<|P| and 2b(P)<|P|.
All hese examples ha e he con o mal ype o C. Once he e, Collin and Rosenbe g
designed a way o enla ging a gi en Sche k- ype g aph o e he in e io o some Γ ⊂H2
in o ano he one wi h mo e sides, and so ha : (1) he ex ended su ace is C2-close o he
o iginal one o e an a bi a y compac se in he in e io o Γ, and (2) he e is a con ol
on he con o mal adius on adequa e compac annuli on he su ace.
By passing o he limi in his sequence o minimal g aphs o e la ge and la ge
domains, hey ob ained an en i e minimal g aph in H2×Rwhich, by he con ol on he
con o mal adii o hese annuli, has he con o mal ype o C.
Rema k 7.4. The Collin-Rosenbe g heo em has been ex ended by J.A. G´al ez and H.
Rosenbe g [GaRo] o mo e gene al p oduc spaces M2×R: he e exis en i e minimal
g aphs o pa abolic con o mal ype on M2×R, whe e (M2, g) is any comple e simply
connec ed Riemannian su ace wi h Gaussian cu a u e KM≤c < 0 (KMno cons an ).
7.2. Minimal su aces o ini e o al cu a u e in H2×R.One
o he mos s udied amilies among minimal su aces in R3a e he comple e minimal
su aces o fini e o al cu a u e (FTC o sho ). A minimal su ace Σ is said o ha e
FTC i i s Gaussian cu a u e Ksa isfies
ZΣ
K dA<∞.
By classical heo ems o Hube and Osse man, comple e FTC minimal su aces in
R3a e con o mally equi alen o a compac Riemann su ace minus a fini e numbe o
poin s. Mo eo e , he Gauss map ex ends me omo phically o he punc u es, and he
o al cu a u e o he su ace is a mul iple o −4π. A key poin he e is ha he Gauss
map o a minimal su ace in R3is con o mal.
Cons an mean cu a u e su aces in 3-dimensional Thu s on geome ies 23
In H2×R he e is no con o mal Gauss map o minimal su aces. None heless, using
he global heo y o ha monic maps in o H2, L. Hauswi h and H. Rosenbe g [HaRo] we e
able o p o e ha a simila si ua ion holds in H2×R.
Theo em 7.5 (Hauswi h-Rosenbe g). Le Xbe a comple e minimal imme sion o Σin
H2×Rwi h fini e o al cu a u e. Then
1. Σis con o mally equi alen o a Riemann su ace punc u ed a a fini e numbe o
poin s, Σ≡Mg−{p1...., pk}.
2. Qdz2:= h2
zdz2is holomo phic on Mand ex ends me omo phically o each punc-
u e. I we pa ame e ize each punc u e piby he ex e io o a disk o adius , and
i Q(z)dz2=z2mi(dz)2a pi, hen mi⩾−1.
3. The hi d coo dina e uo he uni no mal ends o ze o uni o mly a each punc u e.
4. The o al cu a u e is a mul iple o 2π:
ZΣ
KdA = 2π 2−2g−2k−
k
X
i=1
mi!.
As a consequence, e e y end o a fini e o al cu a u e su ace is uni o mly asymp o ic
o a Sche k ype g aph desc ibed in Theo em 7.3.
P oo . The fi s s ep is o p o e ha locally a ound an end, Qdz2only has a mos a fini e
numbe o ze oes. Then a Hube heo em and an a gumen o Osse man gi e ha he ends
a e con o mally a punc u ed disk, and Qdz2ex ends me omo phically o he punc u e.
The final pa o he beha io o Qdz2 ollows om he ac ha Qdz2=h2
zdz2, whe e h
is he heigh unc ion o he su ace.
To p o e ha ugoes o 0 a he ends, ake an annula neighbo hood o an end whe e
Qdz2does no anish. Then epa ame e ize his annulus by w=R√Qdz. The me ic
con o mal ac o in hese coo dina es sa isfies a sinh-Go don equa ion, and he Gaussian
cu a u e mono onically dec eases o ze o. Then, es ima es on he g ow h o solu ions
o he sinh-Go don equa ion allows one o conclude ha , a a fini e o al cu a u e end,
he angen plane becomes e ical and he me ic becomes fla .
Finally, he exp ession o he o al cu a u e ollows om Gauss-Bonne o mula and
he es ima es o he sinh-Go don equa ion ob ained be o e.
In [HaRo], he ollowing ques ion was also aised: a e he e comple e non simply
connec ed minimal su aces wi h FTC in H2×R?No ice ha o a ional ca enoids ha e
infini e o al cu a u e. Ac ually, a ha ime, he only known comple e FTC minimal
su aces we e he Sche k ype g aphs.
This ques ion was posi i ely answe ed by J. Pyo [Pyo] and also, independen ly, by
Rod ´ıguez and Mo abi o [RoMo]. Pyo cons uc ed a 1-pa ame e amily o genus ze o
p ope ly embedded minimal su aces in H2×Rwi h kends o k≥2, simila o he
k-noids in R3(al hough he fi s ones a e embedded and he k-noids in R3a e no ). They
ha e o al cu a u e 4π(1 −k), and a e asymp o ic o e ical planes a infini y. These
su aces a e ob ained as he conjuga e su aces o minimal g aphs o e infini e geodesic
iangles in H2 ha a e asymp o ic o e ical planes a infini y.
Ve y sho ly he ea e , M. Rod ´ıguez and F. Mo abi o disco e ed independen ly a
la ge amily o FTC minimal su aces, con aining he p e ious ones. I is a (2k−
2)-pa ame e o p ope ly embedded FTC minimal su aces o genus ze o wi h kends,
ob ained as he limi s o simply pe iodic minimal su aces called saddle owe s, ha a e
24 Isabel Fe n´andez and Pablo Mi a
in a ian by a e ical ansla ion o ec o (0,0,2l). Taking limi s when l→ ∞, hey
ob ain genus ze o minimal su aces wi h kends and o al cu a u e 4π(1 −k) ha a e
symme ic wi h espec o he eflec ion o e he slice H2×{0}. The su aces ound by
Pyo appea when he ends a e placed in symme ic posi ions.
7.3. Open p oblems. In [Ha], L. Hauswi h cons uc ed a amily o Riemann
ype minimal su aces in H2×Rand S2×R, cha ac e ized by he p ope y o being olia ed
by cu es o cons an cu a u e. I is a conjec u e by W. Meeks and H. Rosenbe g ha
in S2×R hey a e he only p ope ly embedded minimal annuli. An app oach o sol ing
his conjec u e using in eg able sys ems echniques has been ecen ly de eloped by L.
Hauswi h and M. Schmid . Ano he na u al p oblem is o ob ain classifica ion esul s
o p ope ly embedded minimal su aces o fini e o al cu a u e and a gi en simple
opology in R3.
Schoen and Yau p o ed he e is no ha monic diffeomo phism om he disk o a
comple e su ace o non-nega i e cu a u e. Can he e be such a ha monic diffeomo phism
on o a comple e pa abolic su ace? This is a ques ion by J.A. G´al ez.
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