Slab theorem and halfspace theorem for constant mean curvature surfaces in H2 x R

Re . Ma . Ibe oam. 39 (2023), no. 1, 307–320 DOI 10.4171/RMI/1372 © 2022 Real Sociedad Ma emá ica Española Published by EMS P ess and licensed unde a CC BY 4.0 license Slab heo em and hal space heo em o cons an mean cu a u e su aces in H2R Lau en Hauswi h, Ana Menezes and Magdalena Rod íguez Abs ac . We p o e ha a p ope ly embedded annula end o a su ace in H2R wi h cons an mean cu a u e 0<H1=2 can no be con ained in any ho izon- al slab. Mo eo e , we show ha a p ope ly embedded su ace wi h cons an mean cu a u e 0<H1=2 con ained in H2Œ0; C1/and wi h ini e opology is nec- essa ily a g aph o e a simply connec ed domain o H2. Fo he case HD1=2, he g aph is en i e. 1. In oduc ion The heo y o cons an mean cu a u e (CMC) H > 0 su aces in H2Rd ew a lo o a en ion a e he wo k by Ab esch and Rosenbe g [1], whe e hey desc ibed a Hop - ype holomo phic quad a ic di e en ial on any such su ace, and cha ac e ized he CMC o a ional sphe es o H > 1=2 as he only imme sed CMC sphe es in his space [1,12, 18,19]. Fo 0H1=2, he e a e no compac CMC examples. This is why HD1=2 is called he c i ical alue o he mean cu a u e in H2R. The CMC o a ional simply connec ed examples o 0 < H 1=2 a e en i e g aphs o pa aboloid- ype shape (see Sec ion 2.3 o mo e de ails). The geome ic beha io o CMC su aces in H2R o H > 1=2 is, in some sense, analogous o he one o su aces o posi i e CMC in R3. Fo ins ance, o hese alues o he mean cu a u e he e exis sphe es and a 1-pa ame e amily o annuli in a ian by a e ical ansla ion simila o he Delaunay’s examples (see [19] and e e ences he ein). An impo an heo em by Ho man and Meeks [11] in he classical heo y o minimal su aces in R3is he hal space heo em, saying ha he e a e no p ope ly imme sed non- la minimal su aces in a hal space. Howe e , in H2R his esul does no hold: he e a e many en i e minimal g aphs and o a ional annuli (called ca enoids) con ained in a slab cons uc ed by Nelli and Rosenbe g [17]. The e a e also o he p ope ly embedded minimal annuli o bounded heigh cons uc ed in [7]. No ice ha he exis ence o sphe es o H > 1=2 and pa aboloid- ype g aphs o 0 < H 1=2 implies ha en i e g aphs o bounded heigh canno exis in H2R o H > 0 by he maximum p inciple. 2020 Ma hema ics Subjec Classi ica ion: P ima y 53A10; Seconda y 53C30. Keywo ds: Cons an mean cu a u e su ace, slab heo em, hal space heo em. L. Hauswi h, A. Menezes and M. Rod íguez 308 In [4], Collin, Hauswi h and Rosenbe g p o ed ha a p ope ly embedded simply con- nec ed minimal su ace in a slab So heigh less han mus be an en i e g aph. Mo e gene ally, hey p o ed ha each end o a minimal su ace p ope ly embedded in Swi h ini e opology is a g aph ou side a compac domain. Fo he case o CMC su aces wi h 0<H1=2 he beha io is di e en . We p o e ha he e a e no examples in a slab. Theo em (Slab heo em). Le MH2Rbe a su ace (possibly wi h bounda y)wi h cons an mean cu a u e 0<H 1=2 and a leas one p ope ly embedded annula end. Then Mcan no be con ained in a ho izon al slab H2Œ0; L, o any L>0. In pa icula , he e a e no p ope ly embedded CMC su aces o 0<H1=2 wi h ini e opology con ained in a ho izon al slab o H2R. Fo any 0<H1=2, Manzano and To albo cons uc ed in [14] p ope ly imme sed CMC su aces con ained in a slab. These examples a e in a ian by a g oup o symme ies induced by a essella ion o H2by egula polygons. A undamen al domain o any o hese examples is compac and i s li in H2Rhas, by pe iodici y, only one end o in ini e opology. I hese examples we e embedded, i would show ha he hypo hesis o ha ing an annula end in he heo em abo e is sha p. Fo CMC HD1=2, he e exis s a hal space heo em [9] o comple e embedded CMC su aces in H2Rlying on one side o a ho ocylinde (wi h some condi ion on he mean cu a u e ec o ); he only such examples a e pa allel ho ocylinde s. Nelli and Sa Ea p [20] also p o ed ha he only CMC su aces wi h HD1=2 con ained in he mean con ex side o he o a ionally in a ian pa aboloid- ype en i e g aph a e ansla ed copies o he g aph. I we hink o CMC su aces on one side o a ho izon al slice, he only known examples a e en i e g aphs. He e we p o e ha , o 0<H1=2, he only p op- e ly embedded CMC su aces wi h ini e opology con ained in one side o a ho izon al slice a e g aphs. Theo em (Hal space heo em). Le MH2Œ0; C1/be a p ope ly embedded su ace wi h cons an mean cu a u e 0 < H 1=2 and ini e opology. Then Mis necessa ily a g aph o e a simply connec ed domain o H2. Fo HD1=2, he g aph is en i e. 2. P elimina ies In his sec ion we will se up some no a ions and in oduce some classes o cons an mean cu a u e (CMC) g aphs in H2R ha we will use as ba ie s. Th oughou his pape we conside he cylinde model o H2R. We conside H2D ¹.x; y/ 2R2Ix2Cy2< 1º endowed wi h he hype bolic me ic g1D4 .1x2y2/2g0;whe e g0deno es he Euclidean me ic in R2:We will hen conside he uni solid cylinde wi h he p oduc me ic gD g1Cd 2as model o H2R. In his model, he e is a na u al no ion o asymp o ic bounda y o H2Rwhe e .@1H2/RDS1R. 2.1. Cons an mean cu a u e Sche k g aphs Fo any H2.0; 1=2; Hauswi h, Rosenbe g and Sp uck [10] desc ibed necessa y and su icien condi ions o e a compac admissible domain in o de o gua an ee he exis ence Slab heo em and hal space heo em o CMC su aces in H2R309 o a g aph wi h cons an mean cu a u e Hassuming in ini e bounda y alues. A compac domain in H2is said o be admissible i i is simply connec ed and i s bounda y @ is a polygon wi h sides ¹Aiºand ¹Biº, wi h nei he wo consecu i e Aiedges no Biedges, and all sa is ying .Ai/D2H and .Bi/D2H, whe e deno es he geodesic cu a u e wi h espec o he in e io o . The Jenkins–Se in p oblem hey conside ed consis s o inding a solu ion o he equa ion o CMC g aphs in o mean cu a u e H(we will call i H-g aph) which assumes bounda y alues C1 on each Aiand 1 on each Bi. In he case he domain is an ideal quad ila e al wi h wo opposi e edges A1; A2and wo opposi e edges B1; B2, Folha and Melo cons uc ed comple e examples – which a e called comple e Sche k H-g aphs – o any 0 < H < p2=2 (see Appendix in [8]). Since we will use as ba ie s hese comple e Sche k H-g aphs o 0 < H < 1=2, he e we show hei exis ence o his wide ange o he mean cu a u e. P oposi ion 2.1. Fo any 0 < H < 1=2, he e exis s a comple e Sche k H-g aph o e an ideal domain, called Sche k domain, bounded by cu es Aiand Bi(as abo e)wi h .Ai/D2H and .Bi/D 2H; o iD1; 2: P oo . We use a Pla eau conjuga e me hod. Melo [16] cons uc ed a comple e minimal Sche k- ype g aph o e an ideal quad ila e al in any e PSL.2; R/space. Using Daniel’s co espondence [5] and he echniques by Cas o-In an es, Manzano and he hi d au ho in [3], hese g aphs co espond o comple e Sche k H-g aphs in o H2Ras desi ed, wi h 0 < H < 1=2. 2.2. Big aph ho izon al annuli wi h cons an mean cu a u e HD1=2 Fo HD1=2, ins ead o using Sche k- ype g aphs as ba ie s we will use he one-pa ame- e amily o ho izon al annuli ¹Caºa>0 cons uc ed by Daniel and he i s au ho in Sec ion 8 o [6], called ho izon al ca enoids, whose bounda y a in ini y consis s o wo e ical lines (see also [3] o an al e na i e cons uc ion). Up o an isome y, we can assume ha any ho izon al ca enoid Cais symme ic wi h espec o he e ical plane ¹yD0º ha sepa a es bo h ends and wi h espec o he ho izon al plane H2¹0º, and he lowe hal o Cais a g aph which we deno e by Ga(see Figu e 1). Each g aph Gais iso- me ic o a minimal su ace †ain he Heisenbe g space Nil3, called i s conjuga e su ace, and hey sha e he same alues o he angle unc ion D< N; @ >(see Daniel’s co e- spondence in [5]). This minimal su ace †ais a helicoidal su ace bounded by wo e ical geodesics. These bounda y geodesics co espond o he ho izon al cu es a heigh 0o Ga, whe e D0. The pa ame e o he amily ¹Caºa>0 co esponds o he size o he neck o he annuli, whe e by neck we mean he (compac ) in e sec ion cu e be ween he annulus and he e ical plane o symme y ¹yD0º. When agoes o ze o, he limi su ace consis s o he union o wo angen ho ocylin- de s (a pinching is p oduced in his case). The limi su ace become e ical e e ywhe e, e en i we conside di e en ansla ed copies o he annuli, so he limi domain o he domains whe e he g aphs Gaa e de ined is he union o wo angen ho odisks and is oli- a ed by di e gence lines (se s o poin s whe e he g adien o he unc ions a e unbounded), ha a e ho ocycles a he same wo poin s a in ini y. L. Hauswi h, A. Menezes and M. Rod íguez 310 Figu e 1. Hal o an ho izon al ca enoid Cawhich is a g aph Gao e a domain in H2¹0º. Now ansla e he ca enoids so ha any Cais angen o H2¹0ºa he o igin o H2 and his poin is con ained in he neck o he annulus. When adi e ges, he necks o he annuli (all o hem passing h ough he o igin) become as la ge as we wan . Hence, when a goes o C1, he g aphs Gacon e ge o he en i e g aph Igi en explici ly by Sa Ea p in equa ion (31) o [21] which is in a ian by a one-pa ame e amily o hype bolic ansla- ions. No ice ha in Rema k 3.7 o [3] i is p o ed ha he conjuga ed minimal su aces o Gain Nil3con e ge o he minimal en i e g aph in Nil3in a ian by he isome ic ansla ions along a ho izon al geodesic, and Daniel p o ed in Example 5.6 o [5] ha he conjuga e su ace o his en i e minimal g aph is he g aph I. In pa icula , o any d > 0, he e exis s a0such ha Rdis con ained in he domain whe e he g aph Gais de ined o any aa0, whe e RdH2deno es he egion bounded by he wo equidis an cu es a dis ance d o he ho izon al geodesic ¹yD0º  H2, see Figu e 2. Figu e 2. P ojec ion o he g aph Gaand he domains D1and D2o e which Ga ¹ < Mº p ojec s, o la ge M. On he o he hand, he conjuga e su ace †aNil3is olia ed by s aigh lines wi h a Gauss map which is ho izon al a in ini y. Hence, on each s aigh line he unc ion con- e ges uni o mly o ze o a in ini y. This shows ha , ou side a compac se , he uni no mal ec o a any poin o he g aph Gais a bi a ily close o ho izon al. In pa icula , using di e en slide back sequences we can p o e ha , o M > 0 la ge enough, he ansla ion o he su ace Ga ¹ < Mºcon e ges o a ho ocylinde . This uni o m con e gence Slab heo em and hal space heo em o CMC su aces in H2R311 p o es ha he ho izon al cu es Ga ¹ D Mºa e close o wo ho ocycles and con ain any hal equidis an cu es o a geodesic ha ing he same poin s a in ini y (co esponding o he ends o he ho izon al ca enoid). Up o an isome y we can assume ha his geodesic is ¹yD0º  H2. In pa icula , Ga ¹ < Mºis a g aph o e wo unbounded domains D1and D2so ha Rdn.D1[D2/is compac o any d > 0. 2.3. En i e cons an mean cu a u e g aphs in H2R The e is a well-known class o en i e CMC g aphs o any H2.0; 1=2 ha a e o a ion- ally in a ian comple e e ical g aphs wi h emp y asymp o ic bounda y in .@1H2/R (see, o ins ance, [18,19]). We a e going o call hem pa aboloids. One such su ace is he g aph o a con ex unc ion u ha di e ges o C1when app oaching @1H2and akes i s global minimum a heigh 0(up o a e ical ansla ion). We will deno e by PC his su ace h oughou his pape . The symme ic su ace wi h espec o he ho izon al slice a heigh 0will be deno ed by P. When HD1=2,PCis he g aph o he unc ion (in pola coo dina es) u. ; / D1 p1 2;whe e 0 < 1: The mean cu a u e ec o o he pa aboloid PCpoin s upwa ds, and we o ien he su ace by he uni ec o ield NCsuch ha CD hNC; @ iis posi i e. Since PCis o a ionally in a ian , we can hink o Cas a unc ion on 2Œ0; 1/ in pola coo dina es. Mo eo e , since PCis he g aph o a con ex unc ion whose angen plane is becoming e ical a in ini y, Cis s ic ly dec easing and akes all alues in .0; 1. In pa icula , o any ˛2.0; 1/ he e exis k˛; h˛> 0 such ha he egion o PCwhe e C˛coincides wi h PC .H2Œ0;h˛/ and is bounded by a ho izon al ci cle o adius k˛(see Figu e 3). We obse e ha bo h h˛and k˛di e ge as ˛goes o 0. Figu e 3. PC .H2Œ0; h˛/. The pa aboloid PC(as well as P) sepa a es he ambien space in o wo connec ed componen s. We deno e by Pin he mean-con ex componen ( he one whe e he mean cu a u e ec o poin s o), and by Pex he o he one. We will use his no a ion in he p oo o he slab heo em. The e a e many o he examples o comple e CMC g aphs in H2R. In ac , o any H2.0; 1=2 he e a e amilies o CMC g aphs in a ian by a hype bolic o a pa abolic ansla ion, no necessa ily en i e when H < 1=2 (see [21] o he appendix in [15]). The L. Hauswi h, A. Menezes and M. Rod íguez 312 in e sec ion o he asymp o ic bounda y o all hese examples wi h .@1H2/Ris con- ained in one o wo e ical lines, and some o hese examples a e con ained in one side o a ho izon al slice. Mo eo e , he e exis many CMC HD1=2 en i e g aphs which a e no o a ionally in a ian , ob ained as de o ma ion o he pa aboloid, wi h emp y asymp o ic bounda y in .@1H2/R(see [2] o mo e de ails). 3. Slab heo em Collin, Hauswi h and Rosenbe g [4] ha e p o ed ha an annula end o a p ope ly im- me sed minimal su ace con ained in a slab o heigh less han is a mul ig aph ou side a compac domain wi h a ini e numbe o shee s. When 0 < H 1=2, he ollowing heo em p o es he non-exis ence o a p ope ly embedded annula end wi h CMC Hin a slab o any heigh L. We will deno e by L he ho izon al slab H2Œ0; L, o any L > 0. Theo em 3.1. Le MH2Rbe a su ace (possibly wi h bounda y)wi h cons an mean cu a u e 0 < H 1=2 and a leas one p ope ly embedded annula end. Then M can no be con ained in L, o any L>0. In pa icula , he e a e no p ope ly embedded CMC su aces o 0<H1=2 wi h ini e opology con ained in a ho izon al slab o H2R. P oo . Le us suppose, by con adic ion, ha he e exis s one such su ace Mcon ained in L, and call Ea p ope ly embedded annula end o M. Then ELis an annulus wi h compac bounda y @E. Since Eis p ope ly embedded, he e exis s a compac disk D(no necessa ily minimal) wi h @D D@E such ha E[Dis a su ace ha sepa a es H2R in o wo connec ed componen s. Along E, he mean cu a u e ec o E Hdis inguishes hese wo componen s. We call in e io componen he one whe e E Hpoin s o, and ex e io componen he o he one. We deno e by N he uni no mal ec o o Esuch ha E HDHN , and D hN; @ ideno es wha we call he angle unc ion o E. Le us conside a pa aboloid PCwi h he same mean cu a u e 0 < H 1=2 as E(see Sec ion 2.3). I s mean cu a u e ec o is poin ing upwa ds, and we o ien he pa aboloid by he uni ec o ield NCsuch ha CDhNC; @ iis posi i e. Since C akes all alues in .0; 1 and PCis o a ionally in a ian , o any poin p2Ewi h .p/ > 0, we will be able o ansla e PCin such a way ha i passes h ough pwi h NC.p/ DN.p/. In he case .p/ < 0, we ins ead conside a ansla ion o P o ind a pa aboloid wi h he same mean cu a u e ec o as Ea p. Claim 1. The e exis s ˛02.0; 1/ such ha ¹p2EWj.p/j˛0ºcon ains a sequence o di e ging poin s .pn/n2N, i.e., (3.1) d.pn/WD dis H2..pn/; .@E//! C1 and j.pn/j  ˛0; whe e WH2R!H2deno es he ( e ical) p ojec ion on o he i s ac o . I no , o any sequence o poin s pn2Esuch ha d.pnC1/ > d.pn/n( he sequence di e ges ho izon ally), he sequence ¹.pn/ºn2Ncon e ges uni o mly o ze o. We hen Slab heo em and hal space heo em o CMC su aces in H2R313 conside he isome y Tnwhich is he composi ion o a e ical and a ho izon al ansla- ions mapping pn o a ixed poin p0D.0; 0; L=2/ 2H2R, and call EnDTn.E/. I he e is a subsequence o ¹pnºn2Nsuch ha he Gaussian cu a u e o Eis uni- o mly bounded in a small neighbo hood o any pn, hen he e is a subsequence o he su aces Enwhich locally con e ges o a CMC su ace E1whose angle unc ion 1 anishes iden ically in a neighbo hood o p, and E1is con ained in a e ical cylinde o e a comple e cu e o cons an geodesic cu a u e 2H. By he unique con inua ion heo em, he e is a subsequence o ¹Enºn2Ncon e ging o a comple e e ical cylinde , a con adic ion wi h he ac ha all he e ms in he sequence a e con ained in a ixed ho izon al slab o heigh less han 3L (H2ŒL; 2L). Then we can assume ha he cu a u e o Eis no uni o mly bounded in neighbo - hoods o pnand, by passing o a subsequence i necessa y, ha jA.pn/j  n o any n, whe e Adeno es he second undamen al o m o E. We call Bn he connec ed componen ha con ains pno he in e sec ion o Ewi h he ex insic ball B.pn; ı/ cen e ed a pno uni o m small adius ı > 0, and we de ine n.p/ WD d.p; @Bn/jA.p/j; o any p2Bn, whe e ddeno es he ex insic dis ance in H2R. The unc ion n an- ishes on @Bnand n.pn/DıjA.pn/j  ın. We hen deduce ha na ains i s maximum a a poin qn2Bn, and n.qn/ın. On he o he hand, ıjA.qn/j  n.qn/, om whe e we deduce ha jA.qn/j  n. We now conside nWD 1 2d.qn; @Bn/and B0 nBn he connec ed componen o E B.qn; n/ ha con ains qn. Fo any poin q2B0 n, i holds 2 nDd.qn; @Bn/d.qn; q/ Cd.q; @Bn/ nCd.q; @Bn/: Hence, d.q; @Bn/ n. Since 2 njA.qn/j D n.qn/ n.q/  njA.q/j, we conclude ha jA.q/j  2jA.qn/j DW 2n: Since .pn/con e ges uni o mly o ze o, we can assume, by passing o a subsequence i necessa y, ha jj< 1=n in Bn. Thus we ha e ha jj< 1=n and jAj  2nin B0 n. Now we conside a blow up on he me ic go H2Rby a ac o nn; mo e p ecisely, we de ine †nas B0 nwi h he me ic gnDng. We can use he exponen ial map a he poin qn o li he su ace †n o i s angen plane Tqn.H2R/R3, and we ob ain a su ace z †nR3which is a minimal su ace wi h espec o he li ed me ic Qgn; whe e Qgnis he me ic such ha he exponen ial map expqnis an isome y om .z †n;Qgn/ o .†n; gn/. The e o e, z †nBR3.0; n n/and, i z Adeno es he second undamen al o m o z †n, we ha e jz A.0/j D 1and jz A.q/j  2 o all q2z †n: On one hand, we obse e ha n ndi e ges, as 2n nD n.qn/ n.pn/ın. Then he balls BR3.0; n n/con e ge o R3and he me ics gncon e ge o he canonical me ic g0o R3. Fo a ixed n, he compac su aces z †0 kWD z †k BR3.0; n n/o R3, wi h kn, all pass h ough he o igin 0and ha e uni o m bounded cu a u e. Then a subsequence o ¹z †0 kºkcon e ges o a minimal su ace in .R3; g0/passing h ough he o igin 0and jz A.0/j D 1, whe e z Aalso deno es he second undamen al o m o his limi L. Hauswi h, A. Menezes and M. Rod íguez 314 su ace. This a gumen holds o any n, so we can use a diagonal a gumen and ob ain, as a limi o a subsequence o he su aces z †k, a comple e minimal su ace z †in R3wi h 02z †and jz A.0/j D 1. On he o he hand, we knew ha jj< 1=n in B0 n. Then we ob ain ha jj< 1=n in z †n, om whe e we deduce ha he Gauss map o z † akes alues in a neighbo hood o he equa o o S2. Then his limi su ace z †mus be a e ical plane, which con adic s he ac ha jz A.0/j D 1. This p o es Claim 1. Take he sequence ¹pnºn2Ngi en by Claim 1. We can assume, by passing o a subse- quence i necessa y, ha .pn/ > 0 (o .pn/ < 0) o any n. We conside , o each pn, a ansla ion o he pa aboloid PC(o P, depending on he sign o .pn/) angen o E a pnwi h he same mean cu a u e ec o a pn. We deno e such pa aboloid by P.pn/. Since he angle unc ion Co he pa aboloid PCis a dec easing unc ion (by he con exi y o u, see Sec ion 2.3), he e exis s a unique ˇ2.0;˛0/such ha hˇDh˛0CL. Hence, P.pn/ .H2Œ0;L/ is he ansla ion o a subdomain o P˙ .H2Œ0;˙hˇ/, whe e he sign ˙depends on he sign o he angle unc ion a pn. Fo nla ge enough, d.pn/ > 2kˇand he angen pa aboloid P.pn/does no in e sec @E. The local in e sec ion o Eand P.pn/nea pnconsis s o kcu es, wi h k2, mee ing a an equal angle a pn. We deno e by nDE P.pn/ he in e sec ion o he wo su aces. Since he in e sec ion o P.pn/wi h he slab con aining Eis compac , n is also compac . The pa aboloid P.pn/di ides H2Rin o wo componen s: a mean-con ex one, Pin .pn/, and a non mean-con ex one, Pex .pn/. Since he in e sec ion o Pin .pn/wi h a ho izon al slab is compac , any componen o EnP.pn/con ained in Pin .pn/is neces- sa ily compac . Claim 2. The e is no compac componen †o EnP.pn/con ained in Pex .pn/wi h bounda y @† P.pn/. Suppose by con adic ion his is no ue. Then we can ind a e ically ansla ed copy z P.pn/o P.pn/ angen o †a a poin zpsuch ha †is con ained in he mean-con ex side o z P.pn/. By he maximum p inciple, we ge ha he mean cu a u e ec o o † a zppoin s o he non mean-con ex side o z P.pn/. Le .P/be he symme ic copy o P.pn/wi h espec o H2¹0º. We can ansla e .P/so ha i is angen o z P.pn/ (and hence o †) a zp. Then †and .P/sha e he same mean cu a u e ec o a zp and †is con ained in he non mean-con ex side o .P/, a con adic ion by he maximum p inciple, and Claim 2is p o ed. Locally a pn, he se Ennhas a leas ou componen s wi h a leas wo o hem con ained in he mean-con ex componen Pin .pn/. We call †1and †2 wo o hese com- ponen s. Claim 3. †1and †2can be connec ed by an a c E Pin .pn/. Suppose by con adic ion his is no he case. Thus nbounds a leas wo dis inc connec ed componen s R1and R2o EnP.pn/which a e con ained in Pin .pn/whose bounda ies mee a pn. We conside e ical ansla ions TsP.pn/o he pa aboloid, whe e Ts.p/ DpCs@ o s0, which olia e Pin .pn/. Since E Pin .pn/is com- Slab heo em and hal space heo em o CMC su aces in H2R315 pac , he e is a las lea Ts1P.pn/o he olia ion ha mee s R1. Then Ts1P.pn/and R1 a e angen a a poin q1and R1is below Ts1P.pn/. By he maximum p inciple, he mean cu a u e ec o o Eand he mean cu a u e ec o o he pa aboloid Ts1P.pn/a e oppo- si e a q1. The componen R1sepa a es Pin .pn/in o wo connec ed componen s and he mean-con ex one RC 1is compac . Obse e ha R1and R2a e local g aphs nea pn. Since he mean cu a u e ec o o Ea pnis poin ing in o Pin .pn/and Eis embedded, he componen R2is comple ely con ained in RC 1(see Figu e 4) and he compac componen o Pin .pn/nR2is no mean-con ex. We hen each a con adic ion applying he max- imum p inciple wi h he las lea Ts2P.pn/o he olia ion ha mee s R2. This p o es Claim 3. Figu e 4. Connec ed componen s R1and R2o EnP.pn/in Pin .pn/appea ing in he p oo o Claim 3. Le nbe a compac a c in E Pin .pn/linking wo poin s q12†1and q22†2. We can comple e nby a compac segmen 0 i†iwi h endpoin s qiand pnsuch ha ˛nD n[0 1[0 2is a loop in E. I ˛nis homologous o ze o in E, hen i is he bounda y o a disk Dwhich con ains poin s in Pex .pn/close o pn; hence, he disk Dhas a subdomain in Pex .pn/, a con adic ion o Claim 2. This p o es ha ˛nis in he homology class o @E and ˛n[@E bounds a subannulus Ano E. Since we can do his cons uc ion o a sequence o di e ging poin s pn, we can use a ansla ion o he pa aboloid PCso ha PC @E D ; and Anand PCa e angen a a poin . Then by he maximum p inciple we conclude ha , along An, he mean cu a u e ec o o Epoin s in o he solid cylinde AC nbounded by he annulus Anand wo disks wi h bounda ies ˛nand @E. We a e hen in he si ua ion o a mean con ex cylinde wi h bounda ies con ained in wo compac subdomains K.˛n/and K.@E/ o H2R:Since all he cu es ˛na e con ained in he in e sec ion wi h he slab Lwi h ansla ed copies o he mean-con ex side o he same pa aboloid, P˙ .H2Œ0; ˙hˇ/, we can suppose ha he compac se s K.˛n/a e compac balls o uni o m adius. Fi s assume 0 < H < 1=2. Fo nla ge enough, we can suppose ha he e exis s a Sche k domain bounded by a cs A1; A2; B1; B2wi h .Ai/D2H and .Bi/D 2H, whe e deno es he geodesic cu a u e wi h espec o he in e io o , such ha Bisepa a es K.˛n/ om K.@E/, o iD1; 2 (see Figu e 5). In pa icula , we can conclude ha he e ical planes A1Rand A2Rdo no in e sec he annulus An; o he wise we could use hype bolic ansla ions o he CMC H-plane AiR o ge a con adic ion wi h he maximum p inciple (no ice ha he mean cu a u e ec o o An poin s in o he compac egion bounded by i , so we ha e he co ec o ien a ion o apply