Periodic maximal graphs in the Lorentz-Minkowki space L3

P oceedings o XIII Fall Wo kshop on Geome y and Physics Mu cia, Sep embe 20–22, 2004 Publ. de la RSME, Vol. 9 (2005), 149–154 Pe iodic maximal g aphs in he Lo en z-Minkowki space L3 Isabel Fe n´ andez1and F ancisco J. L´ opez2 1[email p o ec ed] 2[email p o ec ed] Depa amen o de Geome ´ıa y Topolog´ıa Uni e sidad de G anada Abs ac . We s udy maximal g aphs in he Lo en z-Minkowski space L3 in a ian unde a disc e e g oup o isome ies and ha ing a fini e numbe o singu- la i ies in i s undamen al piece. We also gi e a me hod o cons uc hem, based on he Weie s ass ep esen a ion o maximal su aces. Keywo ds: Maximal su aces, conelike singula i ies, pe iodic su aces. 2000 Ma hema ics Subjec Classi ica ion: 53C50, 53C42, 53A10 1. In oduc ion Maximal su aces in a Lo en zian mani old a e spacelike su aces wi h ze o mean cu a u e. In he Lo en z-Minkowski space L3 hese su aces a ise as local maxima o he a ea unc ional associa ed o a ia ions o he su ace by spacelike su aces. Also, maximal g aphs in L3a e he solu ions o a quasi-linea ellip ic diffe en ial equa ion, and he e o e a maximum p inciple o hem is sa isfied. As in he case o minimal su aces in he Euclidean space, maximal su aces ha e a con o mal ep esen a ion (Weie s ass ep es- en a ion) in e ms o me omo phic da a on a Riemann su ace. A classical esul by Calabi [1] asse s ha he unique comple e maximal su aces in L3a e he spacelike planes. Howe e , i we allow he exis ence o singula i ies, he e is a as heo y o comple e maximal su aces, see o example [10], [2], [4], [5], [6]. In his pape we ocus ou a en ion on isola ed embedded singula i ies o maximal su aces, also called conelike singula i ies (see [7]). I in adi ion he su ace is comple e o p ope , i u ns ou ha i is a g aph o e any spacelike plane o L3. 150 Pe iodic maximal g aphs in L3 We say ha a su ace is pe iodic i i is in a ian unde a g oup o iso- me ies Go L3ac ing p ope ly and eely on L3.This pape de elops he main esul s ob ained by he au ho s in [3] o pe iodic maximal su aces in he embedded case. In conc e e, we show ha he g oup Gcon ains a fi- ni e index subg oup G0which is a g oup o ansla ions o ank 0 ( ha is, G0={Id}), 1 (singly pe iodic su aces), o 2 (doubly pe iodic su aces). We also use he Weie s ass ep esen a ion o maximal su aces o gi e a ecipe eco e ing hese su aces. Figu e 1: Examples o maximal g aphs wi h isola ed singula i ies 2. P elimina ies 2.1. Spacelike imme sion wi h isola ed singula i ies Th ough his pape L3will deno e he 3-dimensional Lo en z-Minkowski space, ha is L3=(R3,dx 2+dy2−dz2),and Ma diffe en iable su ace. An imme sion X:M−→L3is said o be spacelike i o any p∈M, he angen plane TpMwi h he induced me ic is spacelike, ha is o say, he induced me ic on Mis Riemannian. This me ic induces a con o mal s uc u e on M,andsoi becomesinaRiemannsu ace. Le F⊂Mbe a disc e e closed subse o a diffe en iable su ace Mand ds2a Riemannian me ic in M−F. Take a poin q∈F, an open disk D(q) in Msuch ha D(q)∩F={q}and an iso he mal pa ame e z o ds2on D(q)−{q}.Then w i e ds2=h|dz|2,whe e h(w)>0 o any w∈z(D(q)−{q}). By defini ion, he Riemannian me ic ds2is singula a qi o any disk D(q) and any pa ame e zas abo e, he limi limp→qh(z(p)) anishes (as a ma e o ac , i suffices o check his condi ion jus o one disc and con o mal pa ame e ). The me ic ds2is said o be singula a Fi i is singula a any poin o F. In his case, (M,ds 2) is said o be a Riemannian su ace wi h isola ed singula i ies and Fis he singula se o (M,ds 2). Defini ion 1. Le X:M→L3be a con inuous map. Suppose he e is a disc e e closed F⊂Msubse such ha X|M−Fis a spacelike imme sion and (M,ds 2) is a Riemannian su ace wi h isola ed singula i ies in F, whe e ds2 is he me ic induced by X. Isabel Fe n´ andez and F ancisco J. L´ opez 151 Then, Xis said o be a spacelike imme sion wi h (isola ed) singula i ies a F, and X(M) a spacelike su ace wi h (isola ed) singula i ies a X(F). The ollowing lemma desc ibes he beha io o a spacelike imme sion a ound an isola ed singula i y. Lemma 1 ([3]).Le X:M→L3be a spacelike imme sion wi h isola ed singula i ies and Πa spacelike plane. Label π:L3→Πas he Lo en zian o hogonal p ojec ion. Then, h:= π◦Xis a b anched local homeomo phism and i s b anch poin s co espond o he locally non embedded singula i ies o X. As a consequence, i Xis an embedding locally a ound he singula poin s and is p ope , hen X(M)is a g aph o e any spacelike plane (in pa icula X is an embedding). The same conclusion holds i we eplace p ope by comple e. 2.2. Maximal su aces A maximal imme sion X:M→L3is a spacelike imme sion wi h anishing mean cu a u e. The no ion o maximal imme sion wi h (isola ed) singula i - ies is defined analogously. I X:M→L3is a (e e ywhe e egula ) maximal imme sion, i is known ha he e exis a me omo phic map gwi h |g| = 1 and a holomo phic 1- o m φ3defined on he Riemann su ace Msa is ying ha he ec o ial 1- o m Φ=(φ1,φ 2,φ 3):=(i 2(1 g−g)φ3,−1 2(1 g+g)φ3,φ 3) is holomo phic, non anishing and wi hou eal pe iods in M.Mo eo e , up o a ansla ion Xis gi en by X(p)=Rep p0Φ,whe e p0is an a bi a y poin . Ei he he pai (g,φ3) o he ec o ial 1- o m Φ is called he Weie s ass ep esen a ion o he maximal imme sion X. As men ioned be o e, we will ocus ou a en ion in embedded su aces, and he e o e, we will conside only embedded singula i ies. Fo a mo e gene al ea men o non-embedded maximal su aces wi h isola ed singula i ies see [3]. As a consequence o Lemma 1, any p ope (o comple e) maximal su ace wi h isola ed embedded singula i ies is a g aph o e any spacelike plane. The beha iou o a maximal imme sion a ound embedded isola ed sin- gula i ies is well known (see o example [4], [7]). As a ma e o ac , i D is a disc a ound such a singula i y p, hen D {p}is con o mally equi al- en o an annulus A,and he Gauss map o he imme sion becomes ligh like a he bounda y componen o Aco esponding o he singula i y. Mo eo e , a ound X(p) hesu aceX(M) is asymp o ic o a componen o he ligh cone a X(p).Fo his eason, isola ed embedded singula i ies o maximal su aces a e also called conelike singula i ies. 152 Pe iodic maximal g aphs in L3 Defini ion 2. We say ha a maximal g aph wi h isola ed singula i ies X: M→L3is G-pe iodic i X(M) is in a ian unde a disc e e subg oup Go isome ies ac ing eely and p ope ly o L3.We say ha Xis singly ( esp. doubly) pe iodic i Gis a g oup o ansla ions o ank one ( esp. wo). I in addi ion he quo ien o he singula se o Xunde he ela ion induced by Gis fini e we say ha Xis o fini e ype. Le X:M→L3be a maximal g aph o fini e ype and label F={pα: α∈Λ}⊂Mas i s singula se . Taking in o accoun he local beha io a ound he singula i ies desc ibed abo e and he esul s abou he Koebe uni o miza ion gi en in [8], we can deduce ha he Riemann su ace M F is biholomo phic o a ci cula domain C ∪ α∈ΛDα,whe e Dαa e pai wise disjoin closed discs in Cwhose bounda ies γα:= ∂(Dα),α∈Λ,co espond o he singula i ies. In his se ing, we label M0:= C ∪α∈ΛIn (Dα)(1) and we e e o i as he con o mal suppo o X. The con o mal epa ame e - iza ion X0:C ∪α∈ΛDα→L3ex ends o M0by pu ing X0(γα)=X(pα). 3. Main Resul s Theo em 2 ([3]).Le X:M→L3be a G-pe iodic maximal g aph o fini e ype. Then he subg oup G0o Gconsis ing o he posi i e and o hoch onous ( ha is, p ese ing H2 +)isome ieso G, which is a fini e index subg oup o G, is ei he he iden i y o a g oup o spacelike ansla ions o ank 1 o 2. Ou aim now is o desc ibe he global beha io o he G-pe iodic maximal g aphs o fini e ype when Gis one o he h ee g oups gi en in he abo e heo em in e ms o i s Weie s ass da a. Thus, le X:M→L3be as in he s a emen o he heo em, and conside i s con o mal suppo M0and he con o mal epa ame e iza ion X0:M0→ L3(see Equa ion (1)). Since he isome ies in Gp ese es he singula se we can ega d Gas g oup o ans o ma ions in M0.So, we can conside he induced imme sion ˆ X0:ˆ M0=M0/G →L3/G. I ollows ha ˆ X0is a comple e maximal imme sion wi h a fini e numbe o singula i ies. Mo eo e , since Mis simply connec ed, ˆ X0(ˆ M0) is an embedded su ace in L3/G. I in addi ion Gis a ansla ional g oup, he Weie s ass da a o X0can be also pushed ou o ˆ M0. Obse e ha he Riemann su ace wi h bounda y ˆ M0=M0/G is biho- lomo phic o Σ ∪k j=0In (Dj),whe e Dja e pai wise disjoin closed discs and Isabel Fe n´ andez and F ancisco J. L´ opez 153 Σ=Ci G={Id},Σ=C∗in he singly pe iodic case, and Σ is a o us in he doubly pe iodic case. In o de o use he ools we need o ou pu poses is use ul o wo k wi h bounda yless su aces, o his eason we in oduce he no ion o he double su ace o he con o mal suppo . This su ace is no hing bu he quo ien o ˆ M0∪ˆ M∗ 0by iden i ying hei bounda y componen s, ∂(ˆ M0)≡∂(ˆ M∗ 0),whe e ˆ M∗ 0is he mi o su ace associa ed o ˆ M0(see [9] o mo e de ails). I ollows ha he Weie s ass da a Φ can be ex ended holomo phically o he double su ace Sand sa is y J∗(Φ) = −Φ,whe e J:S→Sis he mi o in olu ion, ha maps each poin o ˆ M0in o i s mi o image and ice e sa (obse e ha he fixed poin se o Jcoincides wi h ∂(ˆ M0)). Theo em 3 ([3]).Le Sbe a compac Riemann su ace o genus k≥0and J:S→Sbe an an iholomo phic in olu ion ha ing k+1 pai wise disjoin Jo dan cu es o fixed poin s γ0,...,γ k.Suppose also ha S ∪k j=0γjhas wo connec ed componen s, namely Ωand J(Ω),any one o hem homeomo phic (and so biholomo phic) o a ci cula domain1in he ex ended complex plane Co in a o us T. Conside a me omo phic ec o ial 1- o m Φ=(φ1,φ 2,φ 3)defined on S, non anishing, wi h J∗(Φ) = −Φand ha ing poles a F∞∪J(F∞),whe e F∞⊂Ωconsis s o one (and in his case he poles a e double) o wo poin s (and in his case he poles a e simple) i Ω⊂Cand F∞=∅i Ω⊂T. Finally, label Gas he g oup o ansla ions o ec o s {Re γΦ:γ∈ H1(Ω0,Z)},whe e Ω0is he quo ien su ace ob ained om Ω F∞by iden i- ying each componen γjo ∂(Ω) o a poin qj/∈Ω,j=0,...,k (qj=qh o j=h). Then, Ghas ank 0 (i F∞has 1 poin ), 1 (i F∞con ains 2 poin s) o 2 (F∞=∅), and he map ˆ X0: Ω F∞→L3/G, ˆ X0=Re(Φ), is well defined and p o ides a comple e maximal su ace wi h k+1 singu- la poin s (namely ˆ X0(γj),j=0,...,k) whose li ing o L3is a G-pe iodic maximal g aph wi h k+1singula poin s in i s undamen al piece. Con e sely, any such su ace can be ob ained in his way. F om he beha iou o he Weie s ass da a desc ibed abo e we can deduce he asymp o ic beha iou o he li ed pe iodic su ace in L3.I u ns ou ha i G={Id} he su ace is asymp o ic a infini y o ei he hal ca enoid o a spacelike plane, and in he singly pe iodic case he su ace is asymp o ic o 1 ha is, an open domain bounded by analy ical ci cles 154 Pe iodic maximal g aphs in L3 wo spacelike hal planes. In he doubly pe iodic case he esul ing su ace is con ained in a slab (see [4], [3] o a de ailed p oo ). Examples o su aces cons uc ed using he abo e ep esen a ion ( o ex- ample, he su aces in Figu e 1) can be ound in [3]. 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